In this paper, the terms chained ternary semigroup, cancellable clement , cancellative ternary
semigroup, A-regular element, π- regular element, π- invertible element, noetherian ternary semigroup are
introduced. It is proved that in a commutative chained ternary semigroup T, i) if P is a prime ideal of T and
x ∉ P then n
n 1
x PT
= P for all odd natural numbers n . ii) T is a semiprimary ternary semigroup. iii) If a ε T is
a semisimple element of T, then < a > w ≠ . iv) If < a >w = 𝜙 for all a ε T, then T has no semisimple
elements. v) T has no regular elements, then for any a ε T, < a >w = 𝜙 or < a >w is a prime ideal. vi) If T is a
commutative chained cancellative ternary semigroup then for every non π-invertible element a, < a >w is either
empty or a prime ideal of T. Further it is proved that if T is a chained ternary semigroup with T\T3= { x } for
some x ε T, then i) T\ { x } is an ideal of T. ii) T = xT1T1 = T1xT1 = T1T1x and T 3 = xTT = TxT = TTx is the
unique maximal ideal of T. iii) If a T and a < x >w then a = xn for some odd natural number n > 1.
iv) T\ < x >w = { x, x 3, x5, . . . . .} or T\< x >w ={x, x 3, . . . , xr} for some odd natural number r. v) If a T
and a < x >w then a = xr for some odd natural number r or a = xn sn tn and sn < x >w or tn < x >w
for every odd natural number n. vi) If T contains cancellable elements then x is cancellable element and < x >w
is either empty or a prime ideal of T. It is also prove that, in a commutative chained ternary semigroup T,
T is archemedian ternary semigroup without idempotent elements if and only if < a >w = for every a T.
Further it is proved that if T is a commutative chained ternary semigroup containing cancellable elements and
< a >w = for every a T , then T is a cancellative ternary semigroup. It is proved that if T is a noetherian
ternary semigroup containing proper ideals then T has a maximal ideal. Finally it is proved that if T is a
commutative ternary semigroup such that T = < x > for some x T, then the following are equivalent.
1) T = {x, x2, x3, ............} is infinite. 2) T is a noetherian cancellative ternary semigroup with x xTT.
3) T is a noetherian cancellative ternary semigroup without idempotents. 4) < a >w = for all a T.
5) < x >w = . and if T is a commutative chained ternary semigroup with T ≠ T 3 , then the following are
equivalent. (1) T={x, x 3, x5, . . . . . . .}, where x T\ T 3 (2) T is Noetherian cancellative ternary semigroup
without idempotents. (3) < a >w = for all a T. Finally, it is proved that If T is a commutative chained
noetherian cancellative ternary semigroup without regular elements, then < a >w = for all a T.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
This document summarizes research on primary ideals in ternary semigroups. It presents 4 theorems:
1) If every ideal in a principally reduced ternary semigroup is primary, the only non-zero prime ideal is the maximal ideal M.
2) If every ideal factors into primary ideals and 0nM=Mn holds, then there are at most 3 principal prime ideals different from M.
3) If a ternary semigroup has exactly 3 primes different from M, it is Noetherian of dimension 3.
4) If primary ideals contain powers of their radicals and prime ideals satisfy a uniqueness property, then either M3=0 or every ideal is
In this paper, the terms, simple ternary Γ-semiring, semi-simple, semisimple ternary Γ-semiring are introduced. It is proved that (1) If T is a left simple ternary Γ-semiringor a lateral simple ternary Γ-semiring or a right simple ternary Γ-semiring then T is a simple ternary Γ-semiring. (2) A ternary Γ-semiring T is simple ternary Γ- semiring if and only if TΓTΓaΓTΓT = T for all a T. (3) A ternary Γ-semiring T is regular then every principal ternary Γ-ideal of T is generated by an idempotent. (4) An element a of a ternary Γ-semiring T is said to be semi simple if a n 1 a a i.e. n 1 a a = <a> for all odd natural number n. (5) Let T be a ternary Γ-semiring and a T . If a is regular, then a is semisimple. (17) a be an element of a ternary Γ-semiringT and a is left regular or lateral regular or right regular, then a is semisimple. (18) Let a be an element of a ternary semiring T and if a is intra regular then a is semisimple
This document discusses various topics related to Fourier series and partial differential equations, including:
- Periodic functions and their properties.
- Fourier series representations of functions over intervals, including the calculation of Fourier coefficients.
- Using Fourier series to solve partial differential equations, including first and second order equations.
- Applications of Fourier series such as image compression using the discrete cosine transform.
This paper focuses on showing that much of the theoretical part of linear algebra works fairly well without determinants and provides proofs for most of the major structure, theorems of linear algebra without resorting to determinants.
Special Elements of a Ternary SemiringIJERA Editor
In this paper we study the notion of some special elements such as identity, zero, absorbing, additive
idempotent, idempotent, multiplicatively sub-idempotent, regular, Intra regular, completely regular, g–regular,
invertible and the ternary semirings such as zero sum free ternary semiring, zero ternary semiring, zero divisor
free ternary semiring, ternary semi-integral domain, semi-subtractive ternary semiring, multiplicative
cancellative ternary semiring, Viterbi ternary semiring, regular ternary semiring, completely ternary semiring
and characterize these ternary semirings.
Mathematics Subject Classification : 16Y30, 16Y99.
This document provides an overview of partial orders and related concepts. It begins with examples of partial orders in real-world contexts like project scheduling. Finite and infinite partial orders are discussed, along with common notations used such as (A, R) to represent a partial order R on a set A. Definitions of reflexive, anti-symmetric, and transitive relations are reviewed. The Hasse diagram is introduced as a tool for visualizing partial orders.
This document contains lecture notes for introductory courses in mathematical logic offered at Penn State University. It covers topics in propositional and predicate calculus, including formulas and connectives, truth assignments, logical equivalence, tableau and proof methods, and completeness and compactness theorems. It is authored by Stephen G. Simpson and copyrighted from 1998-2005.
The International Journal of Engineering and Science (The IJES)theijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
This document summarizes research on primary ideals in ternary semigroups. It presents 4 theorems:
1) If every ideal in a principally reduced ternary semigroup is primary, the only non-zero prime ideal is the maximal ideal M.
2) If every ideal factors into primary ideals and 0nM=Mn holds, then there are at most 3 principal prime ideals different from M.
3) If a ternary semigroup has exactly 3 primes different from M, it is Noetherian of dimension 3.
4) If primary ideals contain powers of their radicals and prime ideals satisfy a uniqueness property, then either M3=0 or every ideal is
In this paper, the terms, simple ternary Γ-semiring, semi-simple, semisimple ternary Γ-semiring are introduced. It is proved that (1) If T is a left simple ternary Γ-semiringor a lateral simple ternary Γ-semiring or a right simple ternary Γ-semiring then T is a simple ternary Γ-semiring. (2) A ternary Γ-semiring T is simple ternary Γ- semiring if and only if TΓTΓaΓTΓT = T for all a T. (3) A ternary Γ-semiring T is regular then every principal ternary Γ-ideal of T is generated by an idempotent. (4) An element a of a ternary Γ-semiring T is said to be semi simple if a n 1 a a i.e. n 1 a a = <a> for all odd natural number n. (5) Let T be a ternary Γ-semiring and a T . If a is regular, then a is semisimple. (17) a be an element of a ternary Γ-semiringT and a is left regular or lateral regular or right regular, then a is semisimple. (18) Let a be an element of a ternary semiring T and if a is intra regular then a is semisimple
This document discusses various topics related to Fourier series and partial differential equations, including:
- Periodic functions and their properties.
- Fourier series representations of functions over intervals, including the calculation of Fourier coefficients.
- Using Fourier series to solve partial differential equations, including first and second order equations.
- Applications of Fourier series such as image compression using the discrete cosine transform.
This paper focuses on showing that much of the theoretical part of linear algebra works fairly well without determinants and provides proofs for most of the major structure, theorems of linear algebra without resorting to determinants.
Special Elements of a Ternary SemiringIJERA Editor
In this paper we study the notion of some special elements such as identity, zero, absorbing, additive
idempotent, idempotent, multiplicatively sub-idempotent, regular, Intra regular, completely regular, g–regular,
invertible and the ternary semirings such as zero sum free ternary semiring, zero ternary semiring, zero divisor
free ternary semiring, ternary semi-integral domain, semi-subtractive ternary semiring, multiplicative
cancellative ternary semiring, Viterbi ternary semiring, regular ternary semiring, completely ternary semiring
and characterize these ternary semirings.
Mathematics Subject Classification : 16Y30, 16Y99.
This document provides an overview of partial orders and related concepts. It begins with examples of partial orders in real-world contexts like project scheduling. Finite and infinite partial orders are discussed, along with common notations used such as (A, R) to represent a partial order R on a set A. Definitions of reflexive, anti-symmetric, and transitive relations are reviewed. The Hasse diagram is introduced as a tool for visualizing partial orders.
This document contains lecture notes for introductory courses in mathematical logic offered at Penn State University. It covers topics in propositional and predicate calculus, including formulas and connectives, truth assignments, logical equivalence, tableau and proof methods, and completeness and compactness theorems. It is authored by Stephen G. Simpson and copyrighted from 1998-2005.
This document is a report on Taylor's Theorem from a mathematics class. It begins with an introduction and objectives. It then defines Taylor's Theorem as giving an approximation of a function around a point using a Taylor polynomial. An example is worked through to approximate e to three decimal places using Taylor's formula. Two activities are presented involving the remainder term in Taylor's formula and applying it to polynomials. The document concludes with an assignment on using Taylor's formula for specific functions and approximating 1/e.
The document discusses Fourier series. A Fourier series expresses a periodic function as an infinite sum of sines and cosines. It is named after Jean-Baptiste Joseph Fourier who made important contributions to studying trigonometric series. A Fourier series breaks down a function into its constituent frequencies and determines the contribution of each frequency to the overall signal. The formula, properties, examples, advantages, and applications of Fourier series are provided.
1. The document discusses the founders and applications of Taylor series and the Runge-Kutta method of numerical analysis. Brook Taylor developed Taylor series and Carl Runge and Martin Kutta developed the fourth order Runge-Kutta method for solving ordinary differential equations.
2. Taylor series and Runge-Kutta methods are used to solve initial value problems for ODEs and have applications in fields like biomechanics, biotechnology, and engineering.
3. Examples are provided to demonstrate the working rules and steps for applying Taylor series and the fourth order Runge-Kutta method to problems.
This document explores limiting the size of topological spaces through cardinal invariants and Arhangel'skii's Theorem. It begins by introducing set theory concepts like cardinals and ordinals. It then discusses topological spaces formed by putting the order topology on ordinals, called ordinal spaces. Finally, it covers cardinal invariants, which place bounds on the size of topological spaces, and proves a particular case of Arhangel'skii's Theorem, which showed that compact, first-countable spaces have at most the cardinality of the reals. The goal is to understand Arhangel'skii's novel "closing off" proof technique for bounding cardinalities of topological spaces.
Fourier series are used to represent periodic functions as the sum of simple oscillating functions like sines and cosines. This allows periodic functions, including discontinuous ones, to be broken down into their constituent frequencies or harmonics. Applications include representing sound waves, light waves, radio signals, and other physical phenomena involving wave motion or vibration. The Fourier coefficients determine the relative importance of each harmonic in the overall signal.
The document discusses the history and development of Taylor series. Some key points:
1) Brook Taylor introduced the general method for constructing Taylor series in 1715, after which they are now named. Taylor series represent functions as infinite sums of terms calculated from derivatives at a single point.
2) Special cases of Taylor series, like the Maclaurin series centered at zero, were explored earlier by mathematicians like Madhava and James Gregory.
3) Taylor series allow functions to be approximated by polynomials and are useful in calculus for differentiation, integration, and approximating solutions to problems in physics.
An Analysis and Study of Iteration Proceduresijtsrd
In computational mathematics, an iterative method is a scientific technique that utilizes an underlying speculation to produce a grouping of improving rough answers for a class of issues, where the n th estimate is gotten from the past ones. A particular execution of an iterative method, including the end criteria, is a calculation of the iterative method. An iterative method is called joined if the relating grouping meets for given starting approximations. A scientifically thorough combination investigation of an iterative method is typically performed notwithstanding, heuristic based iterative methods are additionally normal. This Research provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. Dr. R. B. Singh | Shivani Tomar ""An Analysis and Study of Iteration Procedures"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23715.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/computational-science/23715/an-analysis-and-study-of-iteration-procedures/dr-r-b-singh
The document discusses Fourier series and polynomials. It defines a Fourier polynomial as a finite sum of trigonometric terms with coefficients. Fourier polynomials are periodic functions. Integral formulas are provided for the coefficients of Fourier series. A theorem states that any periodic function can be associated with a Fourier series whose coefficients are given by the integral formulas. Examples show calculating the Fourier series for some simple periodic functions. The discussion is extended to functions that are periodic over an interval of length L rather than just 2π.
This document discusses co-ideals in ternary semigroups. It begins by introducing ternary semigroups and defining co-ideals. It then proves some properties of co-ideals, including that the intersection of two co-ideals is a co-ideal, and the product of two co-ideals is a co-ideal. It also shows that every ideal is a co-ideal. The document then focuses on co-ideals in the specific ternary semigroup of positive integers under multiplication. It proves that a non-empty subset is a co-ideal if and only if it is of the form In, consisting of all integers less than or equal to n
Application of fourier series to differential equationsTarun Gehlot
1. The document discusses the application of Fourier series to solve differential equations of the form y(n) + an-1y(n-1) + ... + a1y' + a0y = f(x), where f(x) is a periodic function.
2. It defines the complex Fourier series representation of a periodic function f(x) and proves relationships between the real and complex Fourier coefficients.
3. A key theorem shows that if f(x) is differentiated, its complex Fourier coefficients are multiplied by ik, which allows applying Fourier techniques to differential equations.
1. The document provides an introduction to Fourier analysis and Fourier series. It discusses how periodic functions can be represented as the sum of infinite trigonometric terms.
2. Examples are given of arbitrary functions being approximated by Fourier series of increasing lengths. As the length of the series increases, the ability to mimic the behavior of the original function also increases.
3. The Fourier transform is introduced as a method to represent functions in terms of sine and cosine terms. It allows problems involving differential equations to be transformed into an algebraic form and then transformed back to find the solution.
1) Fourier series and integrals are used to represent periodic functions as an infinite sum or integral of sines and cosines. They are useful for solving differential equations.
2) The Fourier series of a periodic function f(x) with period T is the sum of its coefficients multiplied by sines and cosines of integer multiples of x/T. The coefficients are calculated using integrals involving f(x).
3) Half range expansions like half range cosines (HRC) and half range sines (HRS) are used when f(x) is defined on a finite interval, by extending it to a periodic function on the entire real line.
This document introduces and studies the concept of ˆ-closed sets in topological spaces. Some key points:
1. ˆ-closed sets are defined as sets whose δ-closure is contained in any semi-open set containing the set.
2. It is shown that ˆ-closed sets lie between δ-closed sets and various other classes like δg-closed and ω-closed sets.
3. Several characterizations of ˆ-closed sets are provided in terms of properties of the difference between the δ-closure of the set and the set itself.
4. The concept of the ˆ-kernel of a set is introduced, defined as the intersection of all ˆ-
This document provides an introduction to partial differential equations (PDEs). It defines PDEs as equations involving an unknown function of two or more variables and certain of its partial derivatives. The document then classifies PDEs as linear, semilinear, quasilinear, or fully nonlinear based on how they depend on the derivatives of the unknown function. It lists many common and important PDEs as examples, including the heat equation, wave equation, Laplace's equation, and Euler's equations. Finally, it outlines strategies for studying PDEs, such as seeking explicit solutions, using functional analysis to prove existence of weak solutions, and developing theories to handle both linear and nonlinear PDEs.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A Comparative Study of Two-Sample t-Test Under Fuzzy Environments Using Trape...inventionjournals
This paper proposes a method for testing hypotheses over two sample t-test under fuzzy environments using trapezoidal fuzzy numbers (tfns.). In fact, trapezoidal fuzzy numbers have many advantages over triangular fuzzy numbers as they have more generalized form. Here, we have approached a new method where trapezoidal fuzzy numbers are defined in terms of alpha level of trapezoidal interval data and based on this approach, the test of hypothesis is performed. Moreover the proposed test is analysed under various types of trapezoidal fuzzy models such as Alpha Cut Interval, Membership Function, Ranking Function, Total Integral Value and Graded Mean Integration Representation. And two numerical examples have been illustrated. Finally a comparative view of all conclusions obtained from various test is given for a concrete comparative study.
Taylor's theorem states that any function satisfying certain conditions can be expressed as a Taylor series. A Taylor series is a series expansion of a function about a point, giving an approximation of the function near that point. The Taylor series for a function y(x) around a point x=x0 is given by y1 = y0 + (x-x0)/1! * y0' + (x-x0)2/2! * y0'' + (x-x0)3/3! * y0''' + ..., providing successive approximations of the function near x0 using derivatives of the function evaluated at x0. Similarly, the Taylor series can be developed around any point x=x1
Maximal Ordered Leftideals in Ordered Ternarysemigroupsirjes
This document summarizes research on maximal ordered leftideals in ordered ternarysemigroups. It begins with introducing ordered ternarysemigroups and related concepts like leftideals. It presents lemmas about properties of ordered leftideals. The main results characterize when every proper ordered leftideal is maximal in terms of the number of proper leftideals. Specifically, every proper leftideal is maximal if there is one unique proper leftideal or two proper leftideals whose union is the ternarysemigroup. Theorems also characterize a maximal leftideal in terms of the complement of the leftideal.
The document discusses parabolas and their key properties. It begins by introducing the standard form of a quadratic function and the steps to find the vertex, line of symmetry, and maximum/minimum value. It then provides examples of completing the square to derive the standard form (y = (x - h)2 + k) from other forms. Using this standard form, the vertex is identified as (h, k), the line of symmetry is defined as x = h, and the maximum/minimum is determined by the sign of the parabola based on whether k represents the highest or lowest point.
Infinite sequence & series 1st lecture Mohsin Ramay
The document is a lecture presentation on computational physics from Dr. Tariq Mahmood at the University of the Punjab. It includes:
- An introduction and qualifications of Dr. Tariq Mahmood as the instructor
- Details about the computational physics course such as classes, assignments, exams, and textbooks
- An overview of the course syllabus covering topics like limits of sequences, infinite series, Taylor and Maclaurin series
- Examples of how infinite sequences and series are used in physics and materials science applications
- Definitions and examples of key concepts like sequences, recursion formulas, convergent and divergent sequences
This document summarizes research on preparing and characterizing thin films of TiO2, SiO2, TiO2-SiO2, and TiO2/SiO2 for potential use in solar cells. The films were deposited on glass substrates using a hydrothermal method. Characterization using XRD, SEM, and UV analysis showed the films had brookite crystal structures with uniform particle sizes around 230-240nm. Optical properties like band gap, refractive index, and extinction coefficient were calculated from UV data. The TiO2-SiO2 and TiO2/SiO2 films showed slightly higher band gaps and lower reflectance, making them suitable as anti-reflection coatings for solar cells. In conclusion, the hydro
Closed Loop Analysis of Single-Inductor Dual-Output Buck Converters with Mix-...IOSR Journals
This document presents an analysis of closed loop operation of single-inductor dual-output buck converters. It begins with an overview of these converters and their applications. It then analyzes the circuit operation in both continuous and discontinuous conduction modes, developing equations for voltage gains, duty cycles, and other parameters. Most importantly, it identifies a new "mix-voltage" operation mode where the input voltage can be lower than one output voltage. Experimental results are presented to validate the analytical equations and this new operating mode.
This document is a report on Taylor's Theorem from a mathematics class. It begins with an introduction and objectives. It then defines Taylor's Theorem as giving an approximation of a function around a point using a Taylor polynomial. An example is worked through to approximate e to three decimal places using Taylor's formula. Two activities are presented involving the remainder term in Taylor's formula and applying it to polynomials. The document concludes with an assignment on using Taylor's formula for specific functions and approximating 1/e.
The document discusses Fourier series. A Fourier series expresses a periodic function as an infinite sum of sines and cosines. It is named after Jean-Baptiste Joseph Fourier who made important contributions to studying trigonometric series. A Fourier series breaks down a function into its constituent frequencies and determines the contribution of each frequency to the overall signal. The formula, properties, examples, advantages, and applications of Fourier series are provided.
1. The document discusses the founders and applications of Taylor series and the Runge-Kutta method of numerical analysis. Brook Taylor developed Taylor series and Carl Runge and Martin Kutta developed the fourth order Runge-Kutta method for solving ordinary differential equations.
2. Taylor series and Runge-Kutta methods are used to solve initial value problems for ODEs and have applications in fields like biomechanics, biotechnology, and engineering.
3. Examples are provided to demonstrate the working rules and steps for applying Taylor series and the fourth order Runge-Kutta method to problems.
This document explores limiting the size of topological spaces through cardinal invariants and Arhangel'skii's Theorem. It begins by introducing set theory concepts like cardinals and ordinals. It then discusses topological spaces formed by putting the order topology on ordinals, called ordinal spaces. Finally, it covers cardinal invariants, which place bounds on the size of topological spaces, and proves a particular case of Arhangel'skii's Theorem, which showed that compact, first-countable spaces have at most the cardinality of the reals. The goal is to understand Arhangel'skii's novel "closing off" proof technique for bounding cardinalities of topological spaces.
Fourier series are used to represent periodic functions as the sum of simple oscillating functions like sines and cosines. This allows periodic functions, including discontinuous ones, to be broken down into their constituent frequencies or harmonics. Applications include representing sound waves, light waves, radio signals, and other physical phenomena involving wave motion or vibration. The Fourier coefficients determine the relative importance of each harmonic in the overall signal.
The document discusses the history and development of Taylor series. Some key points:
1) Brook Taylor introduced the general method for constructing Taylor series in 1715, after which they are now named. Taylor series represent functions as infinite sums of terms calculated from derivatives at a single point.
2) Special cases of Taylor series, like the Maclaurin series centered at zero, were explored earlier by mathematicians like Madhava and James Gregory.
3) Taylor series allow functions to be approximated by polynomials and are useful in calculus for differentiation, integration, and approximating solutions to problems in physics.
An Analysis and Study of Iteration Proceduresijtsrd
In computational mathematics, an iterative method is a scientific technique that utilizes an underlying speculation to produce a grouping of improving rough answers for a class of issues, where the n th estimate is gotten from the past ones. A particular execution of an iterative method, including the end criteria, is a calculation of the iterative method. An iterative method is called joined if the relating grouping meets for given starting approximations. A scientifically thorough combination investigation of an iterative method is typically performed notwithstanding, heuristic based iterative methods are additionally normal. This Research provides a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. Dr. R. B. Singh | Shivani Tomar ""An Analysis and Study of Iteration Procedures"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, URL: https://www.ijtsrd.com/papers/ijtsrd23715.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/computational-science/23715/an-analysis-and-study-of-iteration-procedures/dr-r-b-singh
The document discusses Fourier series and polynomials. It defines a Fourier polynomial as a finite sum of trigonometric terms with coefficients. Fourier polynomials are periodic functions. Integral formulas are provided for the coefficients of Fourier series. A theorem states that any periodic function can be associated with a Fourier series whose coefficients are given by the integral formulas. Examples show calculating the Fourier series for some simple periodic functions. The discussion is extended to functions that are periodic over an interval of length L rather than just 2π.
This document discusses co-ideals in ternary semigroups. It begins by introducing ternary semigroups and defining co-ideals. It then proves some properties of co-ideals, including that the intersection of two co-ideals is a co-ideal, and the product of two co-ideals is a co-ideal. It also shows that every ideal is a co-ideal. The document then focuses on co-ideals in the specific ternary semigroup of positive integers under multiplication. It proves that a non-empty subset is a co-ideal if and only if it is of the form In, consisting of all integers less than or equal to n
Application of fourier series to differential equationsTarun Gehlot
1. The document discusses the application of Fourier series to solve differential equations of the form y(n) + an-1y(n-1) + ... + a1y' + a0y = f(x), where f(x) is a periodic function.
2. It defines the complex Fourier series representation of a periodic function f(x) and proves relationships between the real and complex Fourier coefficients.
3. A key theorem shows that if f(x) is differentiated, its complex Fourier coefficients are multiplied by ik, which allows applying Fourier techniques to differential equations.
1. The document provides an introduction to Fourier analysis and Fourier series. It discusses how periodic functions can be represented as the sum of infinite trigonometric terms.
2. Examples are given of arbitrary functions being approximated by Fourier series of increasing lengths. As the length of the series increases, the ability to mimic the behavior of the original function also increases.
3. The Fourier transform is introduced as a method to represent functions in terms of sine and cosine terms. It allows problems involving differential equations to be transformed into an algebraic form and then transformed back to find the solution.
1) Fourier series and integrals are used to represent periodic functions as an infinite sum or integral of sines and cosines. They are useful for solving differential equations.
2) The Fourier series of a periodic function f(x) with period T is the sum of its coefficients multiplied by sines and cosines of integer multiples of x/T. The coefficients are calculated using integrals involving f(x).
3) Half range expansions like half range cosines (HRC) and half range sines (HRS) are used when f(x) is defined on a finite interval, by extending it to a periodic function on the entire real line.
This document introduces and studies the concept of ˆ-closed sets in topological spaces. Some key points:
1. ˆ-closed sets are defined as sets whose δ-closure is contained in any semi-open set containing the set.
2. It is shown that ˆ-closed sets lie between δ-closed sets and various other classes like δg-closed and ω-closed sets.
3. Several characterizations of ˆ-closed sets are provided in terms of properties of the difference between the δ-closure of the set and the set itself.
4. The concept of the ˆ-kernel of a set is introduced, defined as the intersection of all ˆ-
This document provides an introduction to partial differential equations (PDEs). It defines PDEs as equations involving an unknown function of two or more variables and certain of its partial derivatives. The document then classifies PDEs as linear, semilinear, quasilinear, or fully nonlinear based on how they depend on the derivatives of the unknown function. It lists many common and important PDEs as examples, including the heat equation, wave equation, Laplace's equation, and Euler's equations. Finally, it outlines strategies for studying PDEs, such as seeking explicit solutions, using functional analysis to prove existence of weak solutions, and developing theories to handle both linear and nonlinear PDEs.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A Comparative Study of Two-Sample t-Test Under Fuzzy Environments Using Trape...inventionjournals
This paper proposes a method for testing hypotheses over two sample t-test under fuzzy environments using trapezoidal fuzzy numbers (tfns.). In fact, trapezoidal fuzzy numbers have many advantages over triangular fuzzy numbers as they have more generalized form. Here, we have approached a new method where trapezoidal fuzzy numbers are defined in terms of alpha level of trapezoidal interval data and based on this approach, the test of hypothesis is performed. Moreover the proposed test is analysed under various types of trapezoidal fuzzy models such as Alpha Cut Interval, Membership Function, Ranking Function, Total Integral Value and Graded Mean Integration Representation. And two numerical examples have been illustrated. Finally a comparative view of all conclusions obtained from various test is given for a concrete comparative study.
Taylor's theorem states that any function satisfying certain conditions can be expressed as a Taylor series. A Taylor series is a series expansion of a function about a point, giving an approximation of the function near that point. The Taylor series for a function y(x) around a point x=x0 is given by y1 = y0 + (x-x0)/1! * y0' + (x-x0)2/2! * y0'' + (x-x0)3/3! * y0''' + ..., providing successive approximations of the function near x0 using derivatives of the function evaluated at x0. Similarly, the Taylor series can be developed around any point x=x1
Maximal Ordered Leftideals in Ordered Ternarysemigroupsirjes
This document summarizes research on maximal ordered leftideals in ordered ternarysemigroups. It begins with introducing ordered ternarysemigroups and related concepts like leftideals. It presents lemmas about properties of ordered leftideals. The main results characterize when every proper ordered leftideal is maximal in terms of the number of proper leftideals. Specifically, every proper leftideal is maximal if there is one unique proper leftideal or two proper leftideals whose union is the ternarysemigroup. Theorems also characterize a maximal leftideal in terms of the complement of the leftideal.
The document discusses parabolas and their key properties. It begins by introducing the standard form of a quadratic function and the steps to find the vertex, line of symmetry, and maximum/minimum value. It then provides examples of completing the square to derive the standard form (y = (x - h)2 + k) from other forms. Using this standard form, the vertex is identified as (h, k), the line of symmetry is defined as x = h, and the maximum/minimum is determined by the sign of the parabola based on whether k represents the highest or lowest point.
Infinite sequence & series 1st lecture Mohsin Ramay
The document is a lecture presentation on computational physics from Dr. Tariq Mahmood at the University of the Punjab. It includes:
- An introduction and qualifications of Dr. Tariq Mahmood as the instructor
- Details about the computational physics course such as classes, assignments, exams, and textbooks
- An overview of the course syllabus covering topics like limits of sequences, infinite series, Taylor and Maclaurin series
- Examples of how infinite sequences and series are used in physics and materials science applications
- Definitions and examples of key concepts like sequences, recursion formulas, convergent and divergent sequences
This document summarizes research on preparing and characterizing thin films of TiO2, SiO2, TiO2-SiO2, and TiO2/SiO2 for potential use in solar cells. The films were deposited on glass substrates using a hydrothermal method. Characterization using XRD, SEM, and UV analysis showed the films had brookite crystal structures with uniform particle sizes around 230-240nm. Optical properties like band gap, refractive index, and extinction coefficient were calculated from UV data. The TiO2-SiO2 and TiO2/SiO2 films showed slightly higher band gaps and lower reflectance, making them suitable as anti-reflection coatings for solar cells. In conclusion, the hydro
Closed Loop Analysis of Single-Inductor Dual-Output Buck Converters with Mix-...IOSR Journals
This document presents an analysis of closed loop operation of single-inductor dual-output buck converters. It begins with an overview of these converters and their applications. It then analyzes the circuit operation in both continuous and discontinuous conduction modes, developing equations for voltage gains, duty cycles, and other parameters. Most importantly, it identifies a new "mix-voltage" operation mode where the input voltage can be lower than one output voltage. Experimental results are presented to validate the analytical equations and this new operating mode.
Human Skin Cancer Recognition and Classification by Unified Skin Texture and ...IOSR Journals
This document presents a novel method for automatically segmenting skin lesions in macroscopic images using iterative stochastic region merging based on discrete wavelet transformation. It aims to address challenges like illumination variation, presence of hair, irregular skin color variation, and multiple unhealthy skin regions. The method divides an input image into regions, extracts features like color, texture, skewness and kurtosis, then classifies the image using knowledge-based classification. Experimental results on 60 real images show the proposed method achieves lower segmentation error than level set active contours, skin lesion segmentation, and multidirectional gradient vector flow methods.
Bringing Consistency in the Websites of Higher Educational Institutes (HEIs)...IOSR Journals
This document discusses bringing consistency to the websites of higher education institutions (HEIs) in Pakistan. It begins by outlining the importance of consistency in website design for usability. A study was conducted that involved interviews with stakeholders from HEIs and the Higher Education Commission of Pakistan to identify common features and functions on HEI websites. A questionnaire was also distributed to students and faculty to evaluate consistency across different university websites. The results showed that most HEI websites in Pakistan lack consistency and standardization in their design. The document concludes by proposing a framework for designing consistent and usable HEI websites in Pakistan.
This document compares the performance of three routing protocols for mobile ad hoc networks (MANETs): AOMDV, AOMDV with location information using the DREAM protocol, and AOMDV with energy information. AOMDV is a multipath routing protocol that finds multiple disjoint paths between nodes. AOMDV with DREAM uses location information to route packets more efficiently. AOMDV with energy information selects paths based on the residual energy of nodes to improve energy efficiency and network lifetime. The document provides an overview of these three protocols and reviews previous work comparing their performance based on metrics like energy consumption, packet delivery ratio, and throughput.
Practical Investigation of the Environmental Hazards of Idle Time and Speed o...IOSR Journals
This document presents the results of a study that investigated the environmental hazards of idle time and speed of a compression ignition engine fueled with Iraqi diesel fuel. The study measured emissions of CO, HC, CO2, NOx, particulate matter, and noise from the engine at various idle speeds (900, 1000, 1200, 1500 rpm) over a 20 minute period. The results showed that increasing idle time and decreasing idle speed generally increased emissions and noise, while increasing idle speed improved combustion and reduced most emissions and noise. Specifically, higher idle times and lower speeds increased CO, HC, NOx, PM and noise but decreased CO2, while higher speeds increased CO2 and NOx but decreased other emissions and noise. The aim was
Analytical Review on the Correlation between Ai and NeuroscienceIOSR Journals
This document discusses the relationship between artificial intelligence and neuroscience. It describes how AI has benefited from studying neuroscience to better understand natural intelligence. Specifically, AI has used insights from neuroscience related to learning, perception, and reasoning by modeling neural mechanisms. The document also provides several examples of how AI and robotics have been influenced by neuroscience, including early robots designed to mimic animal behavior and more recent projects that apply insights about the brain to develop artificial neural networks or brain-inspired devices.
This document describes the design of a 16-channel audio mixer. It begins with an introduction to audio mixers and their uses. It then discusses the design methodology, considering factors like the number of input/output channels, power requirements, cost, and portability. The design is divided into several stages: a power stage using a step-down transformer and rectification circuit, a stereo stage for each channel with gain, bass, and treble controls, an auxiliary stage to boost the output signal, and a volume control stage to jointly control the levels. Block diagrams and circuit diagrams are provided to illustrate the design. In conclusion, the 16-channel audio mixer is tested by connecting it to an external amplifier and speakers.
Simulation based Evaluation of a Simple Channel Distribution Scheme for MANETsIOSR Journals
This document presents a proposed multi-channel distribution scheme for mobile ad hoc networks (MANETs) and evaluates it through simulation. The proposed scheme assigns channels to nodes based on their node IDs to avoid control overhead from time synchronization. While neighboring nodes on the same channel is possible, the probability is low given random node distribution. The proposed scheme is compared to a single-channel scheme in ns-2 simulations. Results show the proposed technique has better performance.
Low Power Energy Harvesting & Supercapacitor StorageIOSR Journals
This document describes a system for harvesting energy from human power sources like hand cranking or an exercise bicycle and storing it in a supercapacitor bank. The system includes a hand cranked generator, current limiter, voltage sensing circuit, supercapacitor bank, charge balancing circuit, DC-DC converter, and load. Small amounts of energy generated by human power are stored in the supercapacitor bank, which can then be used to power small electronic devices through the DC-DC converter. The supercapacitors allow for rapid charging and discharging of energy and help utilize intermittent bursts of energy from human power sources.
This document summarizes four different privacy preserving policies for online social networks: Safebook, Lockr, flyByNight, and Persona. Safebook is a decentralized social network that leverages real-life trust between users, with users' data stored and routed by trusted peers. Lockr improves privacy in centralized and decentralized content sharing systems. flyByNight aims to mitigate privacy risks through encrypting information on Facebook. Persona allows users to control who accesses their information. The policies differ in their trust models, ability to form flexible groups, and architectural approaches. Persona is identified as the relatively best method based on these parameters.
This document compares LTE networks using frequency division duplexing (FDD) versus time division duplexing (TDD). FDD uses separate frequencies for downlink and uplink, while TDD uses timesharing of a single frequency between downlink and uplink. TDD can operate with unpaired spectrum and dynamically allocate bandwidth between downlink and uplink. FDD generally provides better support for symmetric traffic like voice calls but requires paired spectrum. The document presents simulation results showing the coverage area and throughput of FDD and TDD LTE networks. It concludes that the preferred duplexing method depends on the intended use and characteristics of the network and traffic.
Android Malware: Study and analysis of malware for privacy leak in ad-hoc net...IOSR Journals
This document discusses analyzing Android malware that can leak privacy information in ad-hoc networks. It proposes using static and dynamic analysis methods to detect malware. In static analysis, reverse engineering is used to detect malicious code by decompiling Android app install files. In dynamic analysis, apps are run in an emulator to monitor their network behavior using tools like Snort. Destinations are then white-listed or blacklisted based on safety. The approach is compared to third party apps and is shown to also be effective at detecting malware that uses internet permissions to leak privacy data in small datasets.
Parallel Hardware Implementation of Convolution using Vedic MathematicsIOSR Journals
This document discusses a parallel hardware implementation of convolution using Vedic mathematics on an FPGA. It aims to improve the speed of convolution by using 16 parallel 4x4 bit Vedic multipliers based on the Urdhva Tiryagbhyam algorithm and optimized adders. The design achieves a delay of 17.996 ns, significantly faster than prior work that used serial processing with one multiplier. Vedic mathematics provides an efficient multiplication approach to serve as the core computation and enable the parallel implementation for faster convolution. The design was coded in VHDL and synthesized on a Xilinx FPGA for verification.
IOSR Journal of Humanities and Social Science is an International Journal edited by International Organization of Scientific Research (IOSR).The Journal provides a common forum where all aspects of humanities and social sciences are presented. IOSR-JHSS publishes original papers, review papers, conceptual framework, analytical and simulation models, case studies, empirical research, technical notes etc.
Improvement of Congestion window and Link utilization of High Speed Protocols...IOSR Journals
This document summarizes a research paper that proposes using a k-nearest neighbors (k-NN) algorithm to help high-speed transport layer protocols like CUBIC better distinguish between packet drops due to network congestion versus other factors like noise. The k-NN algorithm would analyze patterns in packet drop history to classify new drops, helping protocols avoid unnecessary window size reductions when drops are not actually due to congestion. The document provides background on high-speed protocols, issues like underutilization from treating all drops as congestion, and how incorporating k-NN classification could improve protocols' performance in noisy network conditions.
Advancing Statistical Education using Technology and Mobile DevicesIOSR Journals
This document summarizes a study that explored using technology and mobile devices to advance statistical education. The study aimed to evaluate the impact of mobile technology on statistical education and analyze student adoption of mobile technology for learning statistics. It hypothesized that using mobile technology would increase student interest in statistics and that students would be inclined to adopt mobile technology for advanced statistics learning. The study examined how factors like technology acceptance, attitudes towards statistics, user satisfaction, and understanding of statistics concepts related to using an online statistics textbook on computers and iPods.
1. The document discusses cost estimation and management of construction projects. It outlines various factors that must be considered like project planning, scheduling, progress monitoring, time and cost estimation, cost management, change order management, and mitigating cost overruns.
2. Effective cost estimation requires considering construction methodology, resources, duration and feasibility. Time and cost have a strong relationship, as delays increase costs.
3. Cost management ensures a project is completed within budget through accurate budget preparation, crew sequencing, and cost optimization in scheduling. Change order management handles changes fairly.
Determining the Different E-Services Required By the Pakistani CitizensIOSR Journals
This document summarizes a study conducted to determine the different e-services required by citizens in Pakistan. The study found that the top priorities for e-services included:
1. E-education facilities like online courses and access to university information.
2. Online registration of citizens at district and local levels integrated with personal information.
3. Online payment and deduction of taxes.
4. Access to higher government officials and authorities through complaint systems.
5. E-health facilities like online doctor appointments and medical information.
6. Online police complaint services and access to case information.
7. Online training for government employees on IT and internet services.
This document provides a literature review on the effect of using ceramic waste powder in self-compacting concrete. It summarizes 12 research papers that studied properties of self-compacting concrete with additions of various mineral admixtures like fly ash, silica fume, and ceramic waste powder. The papers investigated workability properties like slump flow and passing ability, as well as compressive strength when using these admixtures at different replacement levels of cement. In general, the studies found that moderate additions of ceramic waste powder and other mineral admixtures can improve workability and properties of hardened self-compacting concrete.
P-Pseudo Symmetric Ideals in Ternary Semiringiosrjce
In this paper we introduce and study about pseudo symmetric ideals and P-pseudo symmetric ideals
in ternary semi rings. It is proved that (1) every completely P-Semiprime ideal A in a ternary semi ring T is a Ppseudo
symmetric ideal, (2) If A is a P-pseudo symmetric ideal of a ternary semi ring T then (i) A2 = {x : xn ∈ A
for some odd natural number n∈ N} is a minimal completely P-Semiprime ideal of T, (ii) A4 = {x : < x >n
A
for some odd natural number n} is the minimal P-Semiprime ideal of T containing A, (3) Every P-prime ideal Q
minimal relative to containing a P-pseudo symmetric ideal A in a ternary semi ring T is completely P-prime,
and (4) Let A be an ideal of a ternary semi ring T. Then A is completely P-prime iff A is P-prime and P-pseudo
symmetric. Further we introduced the terms pseudo symmetric ternary semi ring and P-pseudo symmetric
ternary semi ring. It is proved that (1) Every commutative ternary semi ring is a pseudo symmetric ternary semi
ring, (2) Every commutative ternary semi ring is a P-pseudo symmetric ternary semi ring, (3) Every pseudo
commutative ternary semi ring is a P-pseudo symmetric ternary semi ring and (4) If T is a ternary semiring in
which every element is a midunit then T is a P-pseudo symmetric ternary semiring
Weyl's Theorem for Algebraically Totally K - Quasi – Paranormal OperatorsIOSR Journals
This document presents theorems related to algebraically k-quasi-paranormal operators. Specifically, it proves that: (1) Weyl's theorem holds for f(T) where T is algebraically k-quasi-paranormal and f is in the continuous functional calculus; (2) a-Browder's theorem holds for f(S) where S is a perturbation of T and f is in the continuous functional calculus; (3) the spectral mapping theorem holds for the Weyl spectrum of T and the essential approximate point spectrum of T. It provides background, definitions, and lemmas required to prove these results about the spectral properties of algebraically k-quasi-paran
This document provides an overview of statistical inference concepts including:
1. Best unbiased estimators, which have minimum mean squared error for a given parameter. The best unbiased estimator, if it exists, must be a function of a sufficient statistic.
2. Sufficiency and the Rao-Blackwell theorem, which states that conditioning an estimator on a sufficient statistic produces a uniformly better estimator.
3. The Cramér-Rao lower bound, which provides a lower bound on the variance of unbiased estimators. Examples are given to illustrate key concepts like when the bound may not hold.
4. Examples are worked through to find minimum variance unbiased estimators, maximum likelihood estimators, and confidence intervals for various distributions
This document summarizes research on generalized classical Fréchet derivatives in real Banach spaces. It begins by reviewing basic definitions and results for Fréchet derivatives, including the chain rule, mean value theorem, and Taylor's formula. It then proves several theorems on generalized Fréchet derivatives, including a generalized chain rule, mean value theorem, and implicit function theorem. It also presents a generalized Taylor's formula for nth order Fréchet differentiable functions. The proofs of the main results on generalized Fréchet derivatives are provided.
This document summarizes key concepts regarding eigenvalues and eigenvectors of matrices:
- Eigenvalues are scalars such that there exist non-zero eigenvectors satisfying Ax = λx.
- The characteristic equation states that λ is an eigenvalue if and only if it satisfies det(A - λI) = 0.
- A matrix is diagonalizable if it can be written as A = PDP-1, where D is a diagonal matrix of eigenvalues and P is a matrix of corresponding eigenvectors. Diagonalizable matrices can easily compute powers by raising the eigenvalues to powers.
Spectral Continuity: (p, r) - Α P And (p, k) - QIOSR Journals
This document discusses spectral continuity properties for operators belonging to the classes of (p,r)-ΑP (absolute (p,r)-paranormal) operators and (p,k)-Q (quasihyponormal) operators. It is shown that if a sequence of operators from one of these classes converges in norm to an operator T in the same class, then the spectrum, Weyl spectrum, Browder spectrum, and essential surjectivity spectrum are continuous at T. Some key properties used in the proofs are that for these operator classes, the ascent is finite, the single valued extension property is satisfied, and the adjoint satisfies a version of Weyl's theorem.
The document provides an overview of sets and logic. It defines basic set concepts like elements, subsets, unions and intersections. It explains Venn diagrams can be used to represent relationships between sets. Logic is introduced as the study of correct reasoning. Propositions are defined as statements that can be determined as true or false. Logical connectives like conjunction, disjunction and negation are explained through truth tables. Compound statements can be formed using these connectives.
PaperNo14-Habibi-IJMA-n-Tuples and ChaoticityMezban Habibi
This document presents theorems and definitions related to n-tuples of operators on a Frechet space and conditions for chaoticity. It begins with definitions of key concepts such as the orbit of a vector under an n-tuple of operators and what it means for an n-tuple to be hypercyclic or for a vector to be periodic. The main results section presents two theorems, the first characterizing when an n-tuple satisfies the hypercyclicity criterion and the second proving conditions under which an n-tuple of weighted backward shifts is chaotic. The second theorem shows the equivalence of an n-tuple being chaotic, hypercyclic with a non-trivial periodic point, having a non-trivial periodic point, and a
On Generalized Classical Fréchet Derivatives in the Real Banach SpaceBRNSS Publication Hub
This document summarizes research on generalized classical Fréchet derivatives in real Banach spaces. It begins by reviewing basic definitions and results for Fréchet derivatives, including the chain rule, mean value theorem, and Taylor's formula. It then proves that Fréchet derivatives exist and are continuous in real Banach spaces. The main results generalize the chain rule, mean value theorem, and Taylor's formula to higher order Fréchet derivatives in real Banach spaces. Proofs are provided for the generalized chain rule and other theorems.
Fixed Point Theorm In Probabilistic Analysisiosrjce
Probabilistic operator theory is the branch of probabilistic analysis which is concerned with the study of
operator-valued random variables and their properties. The development of a theory of random operators is of
interest in its own right as a probabilistic generalization of (deterministic) operator theory and just as operator
theory is of fundamental importance in the study of operator equations, the development of probabilistic operator
theory is required for the study of various classes of random equations
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
The document discusses fundamental matrices and their properties. A fundamental matrix Ψ(t) is a matrix whose columns are fundamental solutions to the system x' = P(t)x. Ψ(t) satisfies the differential equation Ψ' = P(t)Ψ and is nonsingular. The general solution to the system can be written as x = Ψ(t)c, where c is a constant vector. For an initial value problem, the solution is x = Ψ(t)Ψ-1(t0)x0. The fundamental matrix Φ(t) corresponding to a set of fundamental solutions satisfying initial conditions is also discussed. Matrix exponential functions are introduced as the fundamental matrix
1. The document provides a list of 2 mark questions and answers related to the Signals and Systems subject for the 3rd semester IT students.
2. It includes definitions of key terms like signal, system, different types of signals and their classifications. Properties of Fourier series and Fourier transforms are also covered.
3. The questions address topics ranging from periodic/aperiodic signals, even/odd signals, unit step and impulse functions, Fourier series, Fourier transforms, Laplace transforms, linear and time invariant systems.
The document discusses the wave equation and its application to modeling vibrating strings and wind instruments. It describes how the wave equation can be separated into independent equations for time and position using the assumption that displacement is the product of separate time and position functions. This separation leads to trigonometric solutions that satisfy the boundary conditions of strings fixed at both ends. The solutions represent standing waves with discrete frequencies determined by the length, tension, and density of the string. Similar methods apply to wind instruments with different boundary conditions.
This document defines and provides necessary and sufficient conditions for a pair of operators (T1, T2) on a topological vector space to be syndetically hypercyclic. It begins by introducing key concepts such as hypercyclic pairs and sequences. The main result is that a pair (T1, T2) is syndetically hypercyclic if and only if it satisfies the Hypercyclicity Criterion for syndetic sequences. Additionally, it is shown that if a pair satisfies the Hypercyclicity Criterion for a syndetic sequence, then it is topologically mixing. The proof of the main theorem utilizes the Hypercyclicity Criterion and shows topological weak mixing is equivalent to
1) The lecture discusses the time domain analysis of continuous time linear and time-invariant systems. It covers topics such as impulse response, convolution, and how the output of an LTI system can be determined from its impulse response and the input signal.
2) An example of analyzing the voltage response of an RC circuit to an arbitrary input is presented. The output is the sum of the zero-input response, due to initial conditions, and zero-state response, which is a convolution of the impulse response and input signal.
3) Detectors of high energy photons can be modeled as having an exponential decay impulse response. Examples of characterizing real detectors through measurements of energy resolution, timing resolution, and coincidence point spread
Existence Theory for Second Order Nonlinear Functional Random Differential Eq...IOSR Journals
This document presents an existence theory for solutions to second order nonlinear functional random differential equations in Banach algebras. It begins by introducing the type of random differential equation being studied and defining relevant function spaces. It then states several theorems and lemmas from previous works that will be used to prove the main results. The paper goes on to prove that under certain Lipschitz conditions and boundedness assumptions on the operators defining the equation, the random differential equation has at least one random solution in the given function space. It also shows that the set of such random solutions is compact. The results generalize previous existence theorems to the random case.
The famous Kruskal's tree theorem states that the collection of finite trees labelled over a well quasi order and ordered by homeomorphic embedding, forms a well quasi order. Its intended mathematical meaning is that the collection of finite, connected and acyclic graphs labelled over a well quasi order is a well quasi order when it is ordered by the graph minor relation.
Oppositely, the standard proof(s) shows the property to hold for trees in the Computer Science's sense together with an ad-hoc, inductive notion of embedding. The mathematical result follows as a consequence in a somewhat unsatisfactory way.
In this talk, a variant of the standard proof will be illustrated explaining how the Computer Science and the graph-theoretical statements are strictly coupled, thus explaining why the double statement is justified and necessary.
This document summarizes applications of differential equations to real world systems including cooling/warming, population growth, radioactive decay, electrical circuits, survivability with AIDS, economics, drug distribution in the human body, and a pursuit problem. Examples are provided for each application to illustrate solutions to related differential equations. Key concepts covered include Newton's law of cooling, population models, carbon dating, series circuits, survival models, supply and demand models, compound interest, drug concentration in the body over time, and a mathematical model for a dog chasing a rabbit.
The document discusses the origins and evolution of fuzzy logic, beginning with fuzzy set theory proposed by Zadeh in 1965 which aimed to represent vagueness in natural language using fuzzy sets with non-crisp boundaries. It explains key concepts in fuzzy logic like membership functions, fuzzy set operations, fuzzy relations and compositions. The document also compares classical sets with crisp boundaries to fuzzy sets and contrasts crisp logic with fuzzy logic which allows for degrees of truth between 0 and 1.
Similar to Chained Commutative Ternary Semigroups (20)
This document provides a technical review of secure banking using RSA and AES encryption methodologies. It discusses how RSA and AES are commonly used encryption standards for secure data transmission between ATMs and bank servers. The document first provides background on ATM security measures and risks of attacks. It then reviews related work analyzing encryption techniques. The document proposes using a one-time password in addition to a PIN for ATM authentication. It concludes that implementing encryption standards like RSA and AES can make transactions more secure and build trust in online banking.
This document analyzes the performance of various modulation schemes for achieving energy efficient communication over fading channels in wireless sensor networks. It finds that for long transmission distances, low-order modulations like BPSK are optimal due to their lower SNR requirements. However, as transmission distance decreases, higher-order modulations like 16-QAM and 64-QAM become more optimal since they can transmit more bits per symbol, outweighing their higher SNR needs. Simulations show lifetime extensions up to 550% are possible in short-range networks by using higher-order modulations instead of just BPSK. The optimal modulation depends on transmission distance and balancing the energy used by electronic components versus power amplifiers.
This document provides a review of mobility management techniques in vehicular ad hoc networks (VANETs). It discusses three modes of communication in VANETs: vehicle-to-infrastructure (V2I), vehicle-to-vehicle (V2V), and hybrid vehicle (HV) communication. For each communication mode, different mobility management schemes are required due to their unique characteristics. The document also discusses mobility management challenges in VANETs and outlines some open research issues in improving mobility management for seamless communication in these dynamic networks.
This document provides a review of different techniques for segmenting brain MRI images to detect tumors. It compares the K-means and Fuzzy C-means clustering algorithms. K-means is an exclusive clustering algorithm that groups data points into distinct clusters, while Fuzzy C-means is an overlapping clustering algorithm that allows data points to belong to multiple clusters. The document finds that Fuzzy C-means requires more time for brain tumor detection compared to other methods like hierarchical clustering or K-means. It also reviews related work applying these clustering algorithms to segment brain MRI images.
1) The document simulates and compares the performance of AODV and DSDV routing protocols in a mobile ad hoc network under three conditions: when users are fixed, when users move towards the base station, and when users move away from the base station.
2) The results show that both protocols have higher packet delivery and lower packet loss when users are either fixed or moving towards the base station, since signal strength is better in those scenarios. Performance degrades when users move away from the base station due to weaker signals.
3) AODV generally has better performance than DSDV, with higher throughput and packet delivery rates observed across the different user mobility conditions.
This document describes the design and implementation of 4-bit QPSK and 256-bit QAM modulation techniques using MATLAB. It compares the two techniques based on SNR, BER, and efficiency. The key steps of implementing each technique in MATLAB are outlined, including generating random bits, modulation, adding noise, and measuring BER. Simulation results show scatter plots and eye diagrams of the modulated signals. A table compares the results, showing that 256-bit QAM provides better performance than 4-bit QPSK. The document concludes that QAM modulation is more effective for digital transmission systems.
The document proposes a hybrid technique using Anisotropic Scale Invariant Feature Transform (A-SIFT) and Robust Ensemble Support Vector Machine (RESVM) to accurately identify faces in images. A-SIFT improves upon traditional SIFT by applying anisotropic scaling to extract richer directional keypoints. Keypoints are processed with RESVM and hypothesis testing to increase accuracy above 95% by repeatedly reprocessing images until the threshold is met. The technique was tested on similar and different facial images and achieved better results than SIFT in retrieval time and reduced keypoints.
This document studies the effects of dielectric superstrate thickness on microstrip patch antenna parameters. Three types of probes-fed patch antennas (rectangular, circular, and square) were designed to operate at 2.4 GHz using Arlondiclad 880 substrate. The antennas were tested with and without an Arlondiclad 880 superstrate of varying thicknesses. It was found that adding a superstrate slightly degraded performance by lowering the resonant frequency and increasing return loss and VSWR, while decreasing bandwidth and gain. Specifically, increasing the superstrate thickness or dielectric constant resulted in greater changes to the antenna parameters.
This document describes a wireless environment monitoring system that utilizes soil energy as a sustainable power source for wireless sensors. The system uses a microbial fuel cell to generate electricity from the microbial activity in soil. Two microbial fuel cells were created using different soil types and various additives to produce different current and voltage outputs. An electronic circuit was designed on a printed circuit board with components like a microcontroller and ZigBee transceiver. Sensors for temperature and humidity were connected to the circuit to monitor the environment wirelessly. The system provides a low-cost way to power remote sensors without needing battery replacement and avoids the high costs of wiring a power source.
1) The document proposes a model for a frequency tunable inverted-F antenna that uses ferrite material.
2) The resonant frequency of the antenna can be significantly shifted from 2.41GHz to 3.15GHz, a 31% shift, by increasing the static magnetic field placed on the ferrite material.
3) Altering the permeability of the ferrite allows tuning of the antenna's resonant frequency without changing the physical dimensions, providing flexibility to operate over a wide frequency range.
This document summarizes a research paper that presents a speech enhancement method using stationary wavelet transform. The method first classifies speech into voiced, unvoiced, and silence regions based on short-time energy. It then applies different thresholding techniques to the wavelet coefficients of each region - modified hard thresholding for voiced speech, semi-soft thresholding for unvoiced speech, and setting coefficients to zero for silence. Experimental results using speech from the TIMIT database corrupted with white Gaussian noise at various SNR levels show improved performance over other popular denoising methods.
This document reviews the design of an energy-optimized wireless sensor node that encrypts data for transmission. It discusses how sensing schemes that group nodes into clusters and transmit aggregated data can reduce energy consumption compared to individual node transmissions. The proposed node design calculates the minimum transmission power needed based on received signal strength and uses a periodic sleep/wake cycle to optimize energy when not sensing or transmitting. It aims to encrypt data at both the node and network level to further optimize energy usage for wireless communication.
This document discusses group consumption modes. It analyzes factors that impact group consumption, including external environmental factors like technological developments enabling new forms of online and offline interactions, as well as internal motivational factors at both the group and individual level. The document then proposes that group consumption modes can be divided into four types based on two dimensions: vertical (group relationship intensity) and horizontal (consumption action period). These four types are instrument-oriented, information-oriented, enjoyment-oriented, and relationship-oriented consumption modes. Finally, the document notes that consumption modes are dynamic and can evolve over time.
The document summarizes a study of different microstrip patch antenna configurations with slotted ground planes. Three antenna designs were proposed and their performance evaluated through simulation: a conventional square patch, an elliptical patch, and a star-shaped patch. All antennas were mounted on an FR4 substrate. The effects of adding different slot patterns to the ground plane on resonance frequency, bandwidth, gain and efficiency were analyzed parametrically. Key findings were that reshaping the patch and adding slots increased bandwidth and shifted resonance frequency. The elliptical and star patches in particular performed better than the conventional design. Three antenna configurations were selected for fabrication and measurement based on the simulations: a conventional patch with a slot under the patch, an elliptical patch with slots
1) The document describes a study conducted to improve call drop rates in a GSM network through RF optimization.
2) Drive testing was performed before and after optimization using TEMS software to record network parameters like RxLevel, RxQuality, and events.
3) Analysis found call drops were occurring due to issues like handover failures between sectors, interference from adjacent channels, and overshooting due to antenna tilt.
4) Corrective actions taken included defining neighbors between sectors, adjusting frequencies to reduce interference, and lowering the mechanical tilt of an antenna.
5) Post-optimization drive testing showed improvements in RxLevel, RxQuality, and a reduction in dropped calls.
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Chained Commutative Ternary Semigroups
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 4 (May. - Jun. 2013), PP 49-58
www.iosrjournals.org
www.iosrjournals.org 49 | Page
Chained Commutative Ternary Semigroups
1
G. Hanumanta Rao, 2
A. Anjaneyulu, 3
A. Gangadhara Rao
1
Department of Mathematics, S.V.R.M. College, Nagaram, Guntur (dt) A.P. India.
2,3
Department of Mathematics, V.S.R & N.V.R.College, Tenali, A.P. India.
Abstract : In this paper, the terms chained ternary semigroup, cancellable clement , cancellative ternary
semigroup, A-regular element, π- regular element, π- invertible element, noetherian ternary semigroup are
introduced. It is proved that in a commutative chained ternary semigroup T, i) if P is a prime ideal of T and
x ∉ P then n
n 1
x PT
= P for all odd natural numbers n . ii) T is a semiprimary ternary semigroup. iii) If a ∊ T is
a semisimple element of T, then < a > w
≠ . iv) If < a >w
= 𝜙 for all a ∊ T, then T has no semisimple
elements. v) T has no regular elements, then for any a ∊ T, < a >w
= 𝜙 or < a >w
is a prime ideal. vi) If T is a
commutative chained cancellative ternary semigroup then for every non π-invertible element a, < a >w
is either
empty or a prime ideal of T. Further it is proved that if T is a chained ternary semigroup with TT3= { x } for
some x ∊ T, then i) T { x } is an ideal of T. ii) T = xT1
T1
= T1
xT1
= T1
T1
x and T 3 = xTT = TxT = TTx is the
unique maximal ideal of T. iii) If a T and a < x >w
then a = xn for some odd natural number n > 1.
iv) T < x >w
= { x, x 3, x5, . . . . .} or T< x >w
={x, x 3, . . . , xr} for some odd natural number r. v) If a T
and a < x >w
then a = xr for some odd natural number r or a = xn sn tn and sn < x >w
or tn < x >w
for every odd natural number n. vi) If T contains cancellable elements then x is cancellable element and < x >w
is either empty or a prime ideal of T. It is also prove that, in a commutative chained ternary semigroup T,
T is archemedian ternary semigroup without idempotent elements if and only if < a >w
= for every a T.
Further it is proved that if T is a commutative chained ternary semigroup containing cancellable elements and
< a >w
= for every a T , then T is a cancellative ternary semigroup. It is proved that if T is a noetherian
ternary semigroup containing proper ideals then T has a maximal ideal. Finally it is proved that if T is a
commutative ternary semigroup such that T = < x > for some x T, then the following are equivalent.
1) T = {x, x2, x3, ............} is infinite. 2) T is a noetherian cancellative ternary semigroup with x xTT.
3) T is a noetherian cancellative ternary semigroup without idempotents. 4) < a >w
= for all a T.
5) < x >w
= . and if T is a commutative chained ternary semigroup with T ≠ T 3 , then the following are
equivalent. (1) T={x, x 3, x5, . . . . . . .}, where x T T 3 (2) T is Noetherian cancellative ternary semigroup
without idempotents. (3) < a >w
= for all a T. Finally, it is proved that If T is a commutative chained
noetherian cancellative ternary semigroup without regular elements, then < a >w
= for all a T.
Keywords - chained ternary semigroup, cancellable clement , cancellative ternary semigroup, noetherian
ternary semigroup and ternary group.
I. Introduction :
The algebraic theory of semigroups was widely studied by CLIFFORD and PRESTON [5], [6];
PETRICH [15]. The ideal theory in commutative semigroups was developed by BOURNE [4], HARBANS
LAL [10], SATYANARAYANA [19], [20], MANNEPALLI and NAGORE [14]. The ideal theory in duo
semigroups was developed by ANJANEYULU [1], [2], HOEHNKE [11] and KAR.S and MAITY. B. K[12],
[13]. SANTIAGO [18] developed the theory of ternary semigroups. SARALA. Y, ANJANEYULU. A and
MADHUSUDHANA RAO.D [16], [17] introduced the ideal theory in ternary semigroups and characterize the
properties of ideals. GIRI and WAZALWAR[7] initiated the study of prime radicals in semigroups.
ANJANEYULU. A[1], [2], [3] initiated the study of primary and semiprimary ideals in semigroups. He also
introduced chained duo semigroups. HANUMANTHA RAO.G, ANJANEYULU. A and GANGADHARA
RAO. A[8], [9] introduced the study of primary and semiprimary ideals in ternary semigroups. In this paper we
introduce the notions of chained commutative ternary semigroups, noetherian ternary semigroups and
characterize chained commutative ternary semigroups, noetherian ternary semigroups.
2. Chained Commutative Ternary Semigroups
www.iosrjournals.org 50 | Page
II. Priliminaries :
DEFINITION 2.1 : Let T be a non-empty set. Then T is said to be a ternary semigroup if there exist a mapping
from T×T×T to T which maps ( 1 2 3, ,x x x ) 1 2 3x x x satisfying the condition :
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5x x x x x x x x x x x x x x x ix T, 1 5i .
NOTE 2.2 : For the convenience we write 1 2 3x x x instead of 1 2 3x x x
NOTE 2.3 : Let T be a ternary semigroup. If A,B and C are three subsets of T , we shall denote the set ABC =
: , ,abc a A b B c C .
DEFINITION 2.4 : A ternary semigroup T is said to be commutative provided
abc = bca = cab = bac = cba = acb for all a,b,c T.
DEFINITION 2.5 : A nonempty subset A of a ternary semigroup T is said to be left ternary ideal or left ideal
of T if b, c T, a A implies bca A.
NOTE 2.6 : A nonempty subset A of a ternary semigroup T is a left ideal of T if and only if TTA A.
DEFINITION 2.7 : A nonempty subset of a ternary semigroup T is said to be a lateral ternary ideal or simply
lateral ideal of T if b, c T , a A implies bac A.
NOTE 2.8 : A nonempty subset of A of a ternary semigroup T is a lateral ideal of T if and only if TAT A.
DEFINITION 2.9 : A nonempty subset A of a ternary semigroup T is a right ternary ideal or simply right
ideal of T if b, c T , a A implies abc A
NOTE 2.10 : A nonempty subset A of a ternary semigroup T is a right ideal of T if and only if ATT A.
DEFINITION 2.11 : A nonempty subset A of a ternary semigroup T is a two sided ternary ideal or simply two
sided ideal of T if b, c T , a A implies bca A,
abc A.
NOTE 2.12 : A nonempty subset A of a ternary semigroup T is a two sided ideal of T if and only if it is both a
left ideal and a right ideal of T .
DEFINITION 2.13 : A nonempty subset A of a ternary semigroup T is said to be ternary ideal or simply an
ideal of T if b, c T , a A implies bca A, bac A, abc A.
NOTE 2.14 : A nonempty subset A of a ternary semigroup T is an ideal of T if and only if it is left ideal, lateral
ideal and right ideal of T .
DEFINITION 2.15 : An ideal A of a ternary semigroup T is said to be a proper ideal of T if A ≠ T.
DEFINITION 2.16 : An ideal A of a ternary semigroup T is said to be a trivial ideal provided T A is singleton.
DEFINITION 2.17 : An ideal A of a ternary semigroup T is said to be a maximal ideal provided A is a proper
ideal of T and is not properly contained in any proper ideal of T.
DEFINITION 2.18 : An ideal A of a ternary semigroup T is said to be a principal ideal provided A is an ideal
generated by a for some a T. It is denoted by J (a) (or) < a >.
DEFINITION 2.19 : An ideal A of a ternary semigroup T is said to be a completely prime ideal of T provided
x, y, z T and xyz A implies either x A or y A or z A.
DEFINITION 2.20 : An ideal A of a ternary semigroup T is said to be a prime ideal of T provided X,Y,Z are
ideals of T and XYZ A X A or Y A or Z A.
THEOREM 2.21 : Every completely prime ideal of a ternary semigroup T is a prime ideal of T.
THEOREM 2.22 : Let T be a commutative ternary semigroup . An ideal P of T is a prime ideal if and
only if P is a completely prime ideal.
DEFINITION 2.23 : An ideal A of a ternary semigroup T is said to be a completely semiprime ideal provided
x T, n
x A for some odd natural number n >1 implies x A.
THEOREM 2.24 : An ideal A of a ternary semigroup T is semiprime if and only if X is an ideal of T,
X3
⊆ A implies X ⊆ A.
THEOREM 2.25 : Every prime ideal of a ternary semigroup T is semiprime.
NOTATION 2.26 : If A is an ideal of a ternary semigroup T, then we associate the following four types of sets.
1A = The intersection of all completely prime ideals of T containing A.
2A = {xT: xn
A for some odd natural numbers n}
3A = The intersection of all prime ideals of T containing A.
4A = {xT: n
x A for some odd natural number n}
THEOREM 2.27 : If A is an ideal of a ternary semigroup T, then A 4A 3A 2A 1A .
3. Chained Commutative Ternary Semigroups
www.iosrjournals.org 51 | Page
THEOREM 2.28 : If A is an ideal of a commutative ternary semigroup T, then 1A = 2A = 3A = 4A .
DEFINITION 2.29 : If A is an ideal of a ternary semigroup T , then the intersection of all prime ideals of T
containing A is called prime radical or simply radical of A and it is denoted by A or rad A.
DEFINITION 2.30 : If A is an ideal of a ternary semigroup T , then the intersection of all completely prime
ideals of T containing A is called completely prime radical or simply complete radical of A and it is denoted by
c.rad A.
COROLLARY 2.31 : If a A , then there exist an odd positive integer n such that n
a A.
COROLLARY 2.32 : If A is an ideal of a commutative ternary semigroup T, then
rad A = c.rad A.
DEFINITION 2.33 : An element a of a ternary semigroup T is said to be regular if there exist x, y ∈ T such
that axaya = a.
DEFINITION 2.34 : A ternary semigroup T is said to be regular ternary semigroup provided every element is
regular.
DEFINITION 2.35 : An element a of a ternary semigroup T is said to be left regular if there exist x, y ∈ T
such that a = a3
xy.
DEFINITION 2.36 : An element a of a ternary semigroup T is said to be lateral regular if there exist x, y ∈ T
such that a = xa3
y.
DEFINITION 2.37 : An element a of a ternary semigroup T is said to be right regular if there exist x, y ∈ T
such that a = xya3
.
DEFINITION 2.38 : An element a of a ternary semigroup T is said to be intra regular if there exist x, y ∈ T
such that a = xa5
y.
DEFINITION 2.39 : An element a of a ternary semigroup T is said to be semisimple if a 3
a
i.e. 3
a = < a >.
THEOREM 2.40 : An element a of a ternary semigroup T is said to be semisimple if
a n
a i.e. n
a = < a > for all odd natural number n.
DEFINITION 2.41 : A ternary semigroup T is called semisimple ternary semigroup provided every element in
T is semisimple.
DEFINITION 2.42 : An element a of a ternary semigroup T is said to be an idempotent element provided
3
a a .
THEOREM 2.43 : Let T be a ternary semigroup and a ∊ T. If a is idempotent, then a is semisimple.
THEOREM 2.44 : Let T be a ternary semigroup. If T has no semisimple elements, then T has no
idempotent elements.
DEFINITION 2.45 : A ternary semigroup T is said to be an idempotent ternary semigroup or ternary band
provided every element of T is an idempotent.
THEOREM 2.46 : If T is a ternary semigroup with unity 1 then the union of all proper ideals of T is the
unique maximal ideal of T.
THEOREM 2.47 : If T is a commutative ternary semigroup and A is an ideal of T, then abc A if and
only if <a> <b><c> A.
COROLLARY 2.48 : If T is a commutative ternary semigroup and a, b, c ∊ T, then < abc > = <a>
<b><c>.
DEFINITION 2.49 : An ideal A of a ternary semigroup T is said to be a completely semiprime ideal provided
x T, n
x A for some odd natural number n >1 implies x A.
DEFINITION 2.50 : An ideal A of a ternary semigroup T is said to be semiprime ideal provided X is an ideal
of T and Xn
A for some odd natural number n implies X ⊆ A.
DEFINITION 2.51 : A ternary semigroup T is said to be an archimedean ternary semigroup provided for any
a, b T there exists an odd natural number n such that anTbT.
DEFINITION 2.52 : A ternary semigroup T is said to be a strongly archimedean ternary semigroup provided
for any a, b ∈ T, there exist an odd natural number n such that <a>n
⊆ <b>.
THEOREM 2.53 : Every strongly archimedean ternary semigroup is an archimedean ternary semigroup.
THEOREM 2.54 : If T is a commutative ternary semigroup, then the following are equivalent.
1) T is a strongly archimedean semigroup.
2) T is an archimedean semigroup.
3) T has no proper completely prime ideals.
4) T has no proper prime ideals.
THEOREM 2.55 : An ideal Q of ternary semigroup T is a semiprime ideal of T if and only if Q =Q.
4. Chained Commutative Ternary Semigroups
www.iosrjournals.org 52 | Page
DEFINITION 2.56 : An ideal A of a ternary semigroup T is said to be semiprimary if A is a prime ideal
DEFINITION 2.57 : A ternary semigroup T is said to be semiprimary ternary semigroup if every ideal of T is
a semi primary ideal.
DEFINITION 2.58 : A ternary semigroup T is said to be simple ternary semigroup if T is its only ideal.
THEOREM 2.59 : If T is a left simple ternary semigroup (or) a lateral simple ternary semigroup (or) a
right simple ternary semigroup then T is a simple ternary semigroup.
THEOREM 2.60 : If T is a commutative ternary semigroup such that T3
= T, then every maximal ideal of
T is a prime ideal of T.
THEOREM 2.61 : T is a commutative ternary semigroup such that T3
= Tand T having maximal ideals
then T contains regular elements.
III. Chained Commutative Ternary Semigroups
DEFINITION 3.1 : A ternary semigroup T is said to be a chained ternary semigroup if the ideals in T are
linearly ordered by set inclusion.
NOTE 3.2 : An ideal P of a commutative ternary semigroup T is prime if and only if it is completely prime. i.e.,
P is prime if and only if x, y, z ∊ T , xyz ∊ P ⇒ either x ∊P or y ∊ P or z ∊ P.
NOTATION 3.3 : If A is any ideal of a ternary semigroup T, then denote
w
A =
n
n 1
A
where n is odd natural number.
THEOREM 3.4 : Let T be a commutative chained ternary semigroup and P is a prime ideal of T and
x ∉ P then
n
n 1
x
PT= P for all odd natural numbers n .
Proof : Since x ∉ P and P is prime , xn ∉P for all odd natural numbers n. Since xn ∊ T and P is an ideal of T,
xnPT P for all odd natural numbers n.
Therefore n
n 1
x
PT P for all x ∊ T. Since T is a commutative ternary semigroup, xnT1T1
is an ideal of T.
Since xn ∉ P, xnT1T1
⊈ P. Since T is a chained ternary semigroup, P xnT1T1
for all odd natural numbers n.
Let y ∊ P. Then y ∊ xnT1T1
⇒ y = xn st for some s, t ∊ T1. Now xnst ∊ P, xn ∉ P. Since P is prime, s ∊P or
t ∊P. Therefore y = xn st ∊ xnPT for all odd natural number n and hence P ⊆ xnPT for all odd natural number n.
Hence P ⊆
n
n 1
x
PT for all odd natural numbers n ∊ N. Therefore P =
n
n 1
x
PT.
THEOREM 3.5 : If T is a commutative chained ternary semigroup, then T is a semiprimary ternary
semigroup.
Proof : Let A be an ideal of T. We have A =
n 1
P
= Intersection of all prime ideals of T containing A.
Since T is commutative chained terinary semigroup, we have { Pα : α ∊ △ } forms a chain. By Zorns Lemma,
{ Pα : α ∊ △ } has minimal element say P 𝜷. Therefore A = P 𝜷 and P 𝜷 is a prime ideal of T, and hence
A is prime. Therefore A is a semiprimary ideal of T and hence T is a semiprimary ternary semigroup.
THEOREM 3.6 : Let T be a commutative chained ternary semigroup. If a ∊ T is a semisimple element of
T, then < a > w
≠ .
Proof : Suppose that a is a semisimple element of T. Therefore a ∊ < a >
3
, implies that < a > = < a >
3
.
Therefore a ∊ < a > = < a >n for all odd natural numbers n and hence a ∊
n 1
< a >n
= < a >w
and hence
< a >w
≠ 𝜙 .
COROLLARY 3.7 : Let T be a commutative chained ternary semigroup. If < a >w
= 𝜙 for all a ∊ T, then
T has no semisimple elements.
Proof : Suppose that < a >w
= 𝜙 for all a ∊ T. Suppose if possible T has a semisimple element x. By theorem
3.6, < x >w
≠ . It is a contradiction. Therefore T has no semisimple elements.
COROLLARY 3.8 : Let T be a commutative chained ternary semigroup. If < a >w
= 𝜙 for all a ∊ T, then
T has no idempotent elements.
Proof : Suppose that < a >w
= 𝜙 for all a ∊ T. By theorem 3.7, T has no semisimple elements. By theorem
2.44, T has no idempotent elements.
THEOREM 3.9 : Let T be a commutative ternary semigroup and a ∊ T. Then a is semisimple if and only
if a is left, right, lateral regular and regular.
5. Chained Commutative Ternary Semigroups
www.iosrjournals.org 53 | Page
Proof : Suppose that a is semisimple in T. Therefore a ∊ < a >3
. Since T is commutative,
a ∊ < a >3
= < a3
>. Therefore a = a3
st for some s, t ∊ T. Hence a is left regular.
Since T is commutative, a = a3
st = sa3
t = st a3
= asata. Therefore a is left, right, lateral regular and regular.
Conversely suppose that a is left, right, lateral regular and regular.
Therefore a = a3
st = sa3
t = sta3
= asata for some s, t ∊ T. Now a = a3
st ∊ < a3
>= < a >3
.
Hence a is semisimple.
THEOREM 3.10 : Let T be a chained commutative ternary semigroup. If T has no regular elements, then
for any a ∊ T, < a >w
= 𝜙 or < a >w
is a prime ideal.
Proof : Suppose that T has no idempotent elements and a ∊ T.
We have < a >w
=
n 1
< a >n
.
Assume that < a >w
≠ 𝜙. If possible, suppose that < a >w
is not prime. Then there exist x, y, z ∊ T such that
xyz ∊ < a >w
and x, y, z ∉ < a >w
. By theorem 2.47, < x > < y >< z > = < xyz > ⊆ < a >w
.
Now x, y, z ∉< a >w
, implies that there exists odd natural numbers n, m, p such that x ∉ < a >n
, y ∉ < a >m
and
z ∉ < a >p
.
Consider k = min {n, m, p}. Then x, y, z ∉ < a >k
. Since T is commutative chained ternary semigroup,
we have < a >k
⊆ < x >, < a >k
⊆ < y > and < a >k
⊆ < z >.
Therefore < a > 3k
= < a >k
< a >k
< a >k
⊆ < x > < y >< z > = < xyz > ⊆< a >w
⊆< a >9k
.
Then < a >3k
⊆< a >9k
= < a >3k
< a >3k
< a >3k
and hence a3k ∊
3
3k
a .
Therefore a3k is a semisimple element of T. By theorem 3.9, a3k is a regular element of T.
It is a contradiction. Hence < a >w
is a prime ideal of T.
DEFINITION 3.11 : Let T be ternary semigroup and a ∊ T. Then a is said to be a
left cancellable clement if aax = aay ⇒ x = y,
lateral cancellable clement if axa = aya ⇒ x = y,
right cancellable clement if xaa = yaa ⇒ x = y holds for all x, y ∈ T.
DEFINITION 3.12 : Let T be ternary semigroup and a ∊ T. Then a is said to be cancellable clement if it is
left, lateral and right cancellable element.
DEFINITION 3.13 : A ternary semigroup T is said to be a
left cancellative if abx = aby ⇒ x = y for all a, b ∈ T
lateral cancellative if axb = ayb ⇒ x = y for all a, b ∈ T
right cancellative if xab = yab ⇒ x = y for all a, b ∈ T.
DEFINITION 3.14 : A ternary semigroup T is said to be cancellative ternary semigroup if T is left, lateral and
right cancellative.
THEOREM 3.15 : In a ternary semigroup T, the following are equivalent.
1. T is lateral cancellative.
2. T is left and right cancellative.
3. T is cancellative
Proof : (1) ⇒ (2) : Suppose that ternary semigroup T is lateral cancellative. Therefore axb = ayb ⇒ x = y.
Let a, b, x, y ∈ T such that xab = yab.
Now ab[xab] = ab[yab]] ⇒ a[bxa]b = a[bya]b ⇒ bxa = bya ⇒ x = y.
Thus T is right cancellative. Similarly we can prove that T is left cancellative.
(2) ⇒ (3) : Suppose that ternary semigroup T is left and right cancellative.
Let a, b, x, y ∈ T such that axb = ayb.
Now axb = ayb ⇒ a[axb]b = a[ayb]b ⇒ [aax]bb = [aay]bb⇒ aa[x]bb = aa[y]bb. Since T is left and right
cancellative, we get x = y. Thus T is lateral cancellative.
(3) ⇒ (1) : Suppose that ternary semigroup T is cancellative. By the definition 3.13, T is lateral cancellative.
DEFINITION 3.16 : Let T be a ternary semigroup and a ∊ T. Then a is said to be strongly regular element if
there exists x ∊ T such that axaxa = a.
THEOREM 3.17: Let T be a ternary semigroup and a ∊ T. Then a regular element in T if and only if a is
strongly regular element in T.
Proof : Suppose that a regular element in T. Therefore there exists x, y ∊ T such that axaya = a.
Now axayaxaya = axaya = a ⇒ a(xay)a(xay)a = a. That is asasa = a where (xay) = s. Hence a is strongly
regular.
Conversely, suppose that a is strongly regular element in T. Therefore there exists x ∊ T such that axaxa = a.
Hence a is regular in T.
DEFINITION 3.18 : Let T be a ternary semigroup and a ∊ T. Then a is said to be π- regular if there exists x ∊
T such that an
xan
xan
= an
for some odd natural number n.
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DEFINITION 3.19 : Let T be a ternary semigroup and a ∊ T. Then a is said to be π- invertible element if
there exists x ∊ T such that an
xan
xan
= an
and xan
xan
x = x for some odd natural number n.
THEOREM 3.20 : If T is a commutative chained cancellative ternary semigroup then for every
non π-invertible element a, < a >w
is either empty or a prime ideal of T.
Proof : Suppose that a is a non π-invertible element in T. If < a >w
= 𝜙 then theorem is trivial. Let < a >w
≠ .
If possible, suppose that < a >w
is not prime.
Then there exist x, y, z ∊ T such thatxyz ∊ < a >w
and x, y, z ∉ < a >w
.
By theorem 2.48, < x > < y >< z > = < xyz >. Now x, y, z ∉ < a >w
, implies that there exists odd natural
numbers n, m, p such that x ∉ < a >n
, y ∉< a >m
and z ∉< a >p
.
Consider k = min { n , m, p }. Then x, y, z ∉ < a >k
. Since T is chained ternary semigroup, we have
< a >k
⊆ < x >, < a >k
⊆ < y > and < a >k
⊆ < z >.
Therefore < a > 3k
= < a >k
< a >k
< a >k
⊆ < x > < y >< z > = < xyz > ⊆< a >9k
⊆< a >w
.
Then < a > 3k
⊆< a >9k
=< a > 3k
< a > 3k
< a > 3k
and hence a3k ∊ 3k 3
a . Therefore a 3k is a semisimple
element of T. By theorem 3.9, a 3k is a regular element of T. By theorem 3.17,
a 3k is a strogly regular element of T. Therefore a 3k = a 3kxa 3kxa 3k for some x ∊ T.
Now a 3kxa 3kxa 3k xa 3k= a 3kxa 3k . Since T is cancellative, xa 3kx a 3kx = x. Hence a is a π-invertible
element in T. It is a contradiction. Thus < a >w
is a prime ideal of T.
Hence < a >w
= 𝜙 or < a >w
is prime ideal of T.
THEOREM 3.21 : Let T be a chained ternary semigroup. If T≠T 3 then TT 3 = { x } for some x ∊ T.
Proof : Suppose if possible x, y TT 3 and x ≠ y. Since T is a chained ternary semigroup,
< x > < y > or < y > < x >. If < x > < y >, then x < y > and hence x = yst for some s, t T.
Therefore xT 3, which is not true. If < y > < x >, then y < x > and hence y = xpq for some p, q T.
Therefore y T 3, which is not true. It is a contradiction . Therefore x = y . So there exists unique x T such
that x T 3. Therefore TT 3 = { x }.
THEOREM 3.22 : Let T be a chained ternary semigroup with TT 3 = { x } for some
x ∊ T. Then T { x } is an ideal of T.
Proof : Let a T{x}and s, t T. we have ast T3
. Since x T 3, we have ast ≠ x and hence ast T{ x }.
Hence T{ x } is a right ideal of T. similarly, we can get sta, sat T{ x }. Therefore T { x } is an ideal of T.
THEOREM 3.23 : Let T be a commutative chained ternary semigroup. If T≠T 3 such that TT 3 = { x }
for some x ∊ T, then T = xT1
T1
= T1
xT1
= T1
T1
x and T 3 = xTT = TxT = TTx is the unique maximal
ideal of T.
Proof : Since TT 3 ={ x }, T 3 = T{x}. Now xT1
T1
is an ideal of T and T3
is an ideal of T. Since x T 3
and
T is a chained ternary semigroup, T 3 x T1
T1
. Clearly, xTT T3
. Hence T 3 = TTx = TxT = xTT.
Since T3
is trivial, T 3 = xTT = TxT = TTx is the unique maximal ideal of T.
THEOREM 3.24 : Let T be a commutative chained ternary semigroup with T ≠ T3 such that TT3 = { x
} for some x T. If a T and a < x >w
then a = xn for some odd natural number n > 1
Proof : Since T is a commutative chained ternary semigroup with T ≠ T3 such that TT 3≠ { x }.
By theorem 3.23, T3 = TTx = xTT = T { x }. Since a < x >w
, there exists a odd natural number k such that
a< x >k
. Let n be the least odd positive integer such that a < x >n-2
and a < x >n
.
Therefore a x
n-2
TT xn TT and hence a = x
n-2
st for some s, t T.
If s, t x TT then a = xn snsl
ntntl
n xn TT = < x n
> = < x >n
. It is a contradiction.
Hence s, t x TT. Therefore s = x and t = x. Thus a = xn for some odd natural number n.
If n = 1 then a = x < x >. It is a contradiction. Therefore n > 1.
THEOREM 3.25 : Let T be a commutative chained ternary semigroup with TT 3 = { x }. Then T < x >w
= { x, x 3, x5, . . . . .} or T< x >w
={x, x 3, . . . , xr} for some odd natural number r.
Proof : By theorem 3.24, T< x >w
{ x, x 3, x5, . . . ..}. If xn T< x >w
for all odd natural number n, then
T < x >w
= { x, x 3, x5, . . . . } . If xn T < x >w
for some odd natural number n, then we can choose the least
odd positive integer r is such that x
r+2
T < x >w
. Therefore x, x
3
, . . x
r
T < x >w
for all n > r.
Therefore T < x >w
={ x, x 3, x5, . . . ., xr}.
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THEOREM 3.26 : Let T be a commutative chained ternary semigroup with T ≠ T3 such that TT 3 = { x
}. If a T and a < x >w
then a = xr for some odd natural number r or a = xn sn tn and sn < x >w
and tn < x >w
for every odd natural number n or a = xmz where z < x >w
for some even natural
number m.
Proof : Since T is a commutative chained ternary semigroup with T ≠ T 3 such that x TT 3. By theorem
3.23, T3 = TTx = xTT = T { x }.
Let a T. Suppose that a < x >w
. Now a < x >w
implies that a n
n
x
1
.
Therefore a < x >n
= < x n
> for every odd natural number n . Therefore a = xnsn tn for some s
n
, t
n
T for
every odd natural number n.
Case 1 : If s
n
, t
n
< x >w
for some odd natural number n. By theorem 3.24, s
n
= xr, t
n
= xp for some odd natural
number r, p >1 and hence a = xn+r+p for some odd natural number n+ r + p.
Case 2 : If sn, tn < x >w
, then a = xnsn tn where sn , tn < x >w
.
Case 3 : If only one of the sn or tn < x >w
. Suppose that sn < x >w
and tn < x >w
then tn = xp for some
odd natural number p. Therefore a = xnsn tn = a = xn+p sn where n + p is even.
Hence a = xmz where z < x >w
for some even natural number m.
THEOREM 3.27 : Let T be a commutative chained ternary semigroup with TT 3 = { x }. If T contains
cancellable elements then x is cancellable element and < x >w
is either empty or a prime ideal of T.
Proof : Suppose , if possible x is not cancellable in T. Let Z be the set of all non cancellable elements of T.
Clearly x Z. So Z is non empty subset of T. Let a Z and s, t T.
Since a Z, a is not cancellable in T. So there exists b, c T such that aab = aac and b ≠ c.
Now aab = aac stst(aab) = stst(aac) (sta) (sta) b = (sta) (sta)c and b ≠ c.
Hence sta Z and hence Z is a left ideal of T. Since T is a commutative ternary semigroup, Z is an ideal of T.
Since T T 3 = { x }, by theorem 3.23, T = x T1 T1. Since x Z, Z is an ideal of T,
x T1 T1 Z . Thus T Z and hence T = Z. Therefore every element of T is non cancellable. It is a
contradiction. Therefore x is cancellable element in T.
Suppose that < x >w
≠ . Let a, b, c T such that abc< x >w
.
Suppose if possible a < x >w
, b < x >w
and c < x >w
. Now a, b, c < x >w
, implies that by theorem 3.24,
a = xn, b = xm and c = xp for some odd natural numbers n, m, p.
Therefore xn+m+p = abc < x >w
< x >n+m+p+2
, implies that xn+m+p = xn+m+p+2st for some s, t ∊ T.
Now xn+m+p = xn+m+p+2st and x is cancellative, implies that x = x3st for some s, t T.
Therefore x = x3st T 3. It is a contradiction. Therefore either a < x >w
or b < x >w
or
c < x >w
and hence < x >w
is a prime ideal. Therefore < x >w
is either empty or a prime ideal of T.
THEOREM 3.28 : Let T be a commutative chained ternary semigroup. Then T is archemedian ternary
semigroup without idempotent elements if and only if < a >w
= for every a T.
Proof : Suppose that T is an archemedian ternary semigroup without idempotents. If possible, suppose that
< a >w
for some a T. By theorem 3.10, < a >w
is a prime ideal of T. Since T is an archemedian
commutative ternary semigroup, by theorem 2.54, T has no proper prime ideals. Therefore < a >w
= T.
Now a < a >w
< a > 3
and hence a is semisimple. By theorem 3.9, a is regular. So T has idempotent
elements. It is a contradiction. Hence < a >w
= for every a S. Conversely suppose that < a >w
= for every
aS. Since < a >w
= for every a T, By corollary 3.7, T has no semisimple elements. By theorem 2.44,
T has no idempotent elements. If possible, suppose that P is proper prime ideal of T.
Let x T such that x P. Since x P, by theorem 3.3, P =
1n
xn
PT < x >w
. Therefore P < x >w
= . It is
a contradiction. Hence T has no proper prime ideals. By theorem 2.54, T is an archemedian ternary semigroup.
THEOREM 3.29 : If T is a commutative chained ternary semigroup containing cancellable elements and
< a >w
= for every a T, then T is a cancellative ternary semigroup.
Proof : Let T be a commutative chained ternary semigroup containing cancellable elements. Suppose that
< a >w
= for every a T.
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Let Z be the set of all noncancellative elements in T. If possible, suppose that Z is a nonempty subset of T.
If x Z, then there exists y, z T such that xxy = xxz and y z. Therefore for any s, t T, stst(xxy) = stst(xxz)
implies that (stx) (stx)y = (stx)(stx) z and y z. Hence (stx) Z. Therefore Z is a left ideal of T and T is
commutative, implies that Z is an ideal of T.
If possible, suppose that Z is not prime. Then there exists a, b, c T such that abc Z and a, b, c Z. Now
abc Z, implies that (abc)(abc)x = (abc) (abc)y for some x, y T and x y. Hence aa(bc) (bc)x =
aa(bc)(bc)y and a Z, implies that (bc)(bc)x = (bc)(bc)y. Similarly b, c Z, x = y . It is a contradiction.
Therefore Z is a prime ideal of T. Since < a >w
= for every a T, by theorem3.28, we have T is an
archemedian ternary semigroup without idempotents. Therefore by theorem 2.54, T has no proper prime ideals
and hence Z = T.
It is contradiction to T contains cancellable elements. Hence Z = . Thus T is cancellative ternary semigroup.
DEFINITION 3.30 : A ternary semigroup T is said to be ternary group if for all a, b, c ∊ T, there exists
x, y, z ∊ G such that [xab] = [ayb] = [abz] = c.
THEOREM 3. 31 :If T is a ternary semigroup and a, b ∊ T, then abT = { abt : t ∊ T } is a right ideal of T.
Proof : Let x ∊ abT and s, t ∊ T. Now x ∊ abT, implies that x = abu for some u ∊T.
Since s, t, u ∊ T, we have ust ∊ T. Therefore abust ∊ abT . That is xst ∊ abT. Hence abT is a right ideal of
T.
COROLLARY 3.32 : If T is a ternary semigroup and a, b ∊ T, then Tab = { tab : t ∊ T } is a left ideal of T.
Proof : The proof of the theorem follows the above theorem.
COROLLARY 3.33 : If T is a commutative ternary semigroup and a, b ∊ T, then
Tab = { tab : t ∊ T } is an ideal of T.
COROLLARY 3.34 : If T is a commutative ternary semigroup and a, b ∊ T, then
abT = { abt : t ∊ T } is an ideal of T.
COROLLARY 3.35 : If T is a commutative ternary semigroup and a, b ∊ T, then
aTb = { atb : t ∊ T } is an ideal of T.
COROLLARY 3.36 : If T is a ternary group and a ∊ T, then Taa = { taa : t ∊ T } is a left ideal of T.
COROLLARY 3.37 : If T is a ternary group and a ∊ T, then aaT = { aat : t ∊ T } is a left ideal of T.
THEOREM 3.38 : Let T be a commutative chained ternary semigroup. Then T is ternary group if and
only if T is simple ternary semigroup.
Proof : Suppose that T is a ternary group. Let A be an ideal of T. Clearly, A ⊆ T. Let t ∊ T and a ∊ A.
Now t, a ∊ T and T is ternary group, implies that the equation axa = t has solution in T.
Therefore there exists s ∊T, such that asa = t. Hence t = asa ∊ < a > ⊆ A. Therefore A = T.
Thus T has no proper ideals. Hence T is simple ternary group.
Conversely, suppose that T is simple ternary semigroup. Therefore T has no proper ideals. Let a, b, c ∊ T.
By theorem 3.31, we have abT = {abt : t ∊ T} is a right ideal of T. Since T is commutative, abT is an ideal of
T. Since T has no proper ideals, we have abT = T. Therefore c ∊ T = abT. Therefore, there exists s ∊ T, such
that c = abs. Hence the equation abx = c has a solution in T. Similarly, we can prove the equations axb = c and
xab = c has solution in T. Thus T is a ternery group.
COROLLARY 3.39 : If T is a commutative ternary group, then abT = Tab = aTb = T for all a, b ∊ T.
COROLLARY 3.40 : If T is a commutative ternary group, then aaT = Taa = aTa = T for all a ∊ T.
THEOREM 3.41 : If T is a ternary group, then every element of T is regular element in T.
Proof : Suppose that T is a ternary group and a ∊ T. By corollary 3.40, we have aTa = T. Now a ∊ T and aTa =
T, implies that a ∊ aTa. Therefore, a = axa for some x ∊ T.
Hence axaxa = axa = a. Therefore a is strongly regular and hence regular in T. Thus every element of T is
regular element in T.
THEOREM 3.42 : If T is a commutative cancellative archemedian chained ternary semigroup with
< a >w
for some a T, then T is a ternary group.
Proof : Let T be a commutative cancelative archemedian chained ternary semigroup with
< a >w
for some a T. If possible, suppose that T has no idempotent elements.
Since < a >w
, then by theorem 3.10, < a >w
is a prime ideal of T. Since T is an archemedian commutative
ternary semigroup by theorem 2.53, T has no proper prime ideals. It is a contradiction. Hence T has idempotent
elements. Let e be an idempotent element in T. Then xe 3 = xe for every x T. Since T is cancellative , we
have xee = x for every x ∊ T. Since T is commutative, eex = exe = xee = x for every x T. Let a, b, c T.
Now e, b, a T and T is archemedian ternary semigroup, implies that en < a > and en < b > for some odd
natural number n. Since T is commutative, e aTT and e TTb. Therefore e = axy and e = pqb for some
x, y, p, q T. Now c = ece = (axy)c(pqb), implies that c = a(xycpq)b. Therefore s = xycpq is the solution of
9. Chained Commutative Ternary Semigroups
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c = asb. Since T is commutative, the equations axb = abx = xab = c has solution in T.
Therefore T is a ternary group.
DEFINITION 3.43 : A ternary semigroup T is said to be a noetherian ternary semigroup if every ascending
chain of ideals becomes stationary ; i.e., If A1 A 3 A3 ....... is an ascending chain of ideals of T,
then there exists a odd natural number m such that Am = An for all natural numbers n m.
THEOREM 3.44 : If T is a noetherian ternary semigroup containing proper ideals then T has a maximal
ideal.
Proof : Let A1 be a proper ideal of T. If A1 is not a maximal ideal, then there exists a proper ideal A2
of T
such that A1
A2
. If A2
is not a maximal ideal, then there exists a proper ideal A3
of T such that
A1
A2
A3
. By continuing this process we get an ascending chain of proper ideals of T. Since T is
noetherian, The chain A1
A2
A3
. . . . is stationary. Therefore there exists a odd natural number n such
that An = An+1
= An+2
= . . . . Therefore An is maximal ideal of T. Hence T has a maximal ideal.
THEOREM 3.45 : If T is a commutative ternary semigroup such that T = < x > for some xS, then the
following are equivalent.
1) T = {x, x3, x5, . . . . } is infinite.
2) T is a noetherian cancellative ternary semigroup with x xTT.
3) T is a noetherian cancellative ternary semigroup without regular elements.
4) < a >w
= for all a T.
5) < x >w
= .
Proof : (1) (2) : Suppose that T = {x, x3, x5, .....} is infinite. Therefore T = < x > and x T.
Therefore every ideal of T principle ideal of T. Let A1 A2 A3 ....... be an ascending chain of ideals of T.
Therefore A =
1i
Ai also an ideal of T and A is a principle ideal of T. Suppose that A = < a > for some a T.
Then a
1i
Ai and hence aAt for some odd natural number t.
Therefore A =
1i
Ai = < a > At
1i
Ai, and hence A =
1i
Ai = At.
Therefore At = At+1 = At+3 =....... and hence T is a noetherian ternary semigroup.
Let a, b, c, d S such that abc = abd. Now a, b, c, d S = < x > imples that a = xn, b = xm, c = xs , d = xp for
some odd natural numbers n, m, s, p N. Now abc = abd, implies that
xn xm xs =xn xmxp ⇒ xn+m+s = xnm+ps. Since T is infinite set, n + m + s = n + m + p and hence
s = p. Therefore xs = xp ⇒ b = c. Hence T is cancellative. Suppose that x xTT.
Therefore x = xxnxm for some odd natural numbers n, m. Thus xn+m+1 = x and hence T is finite. It is a
contradiction. So x xTT. Therefore T is a noetherian, cancellative ternary semigroup and x xTT.
(2) (3) : Suppose that T is a noetherian cancellative ternary semigroup and x xTT. If possible, suppose that
T has idempotent elements. Let e be an idempotent element in T. Therefore xe 3e= xee. Thus (xe2
) ee = xee ⇒
xe2 = x. Since T is a cancellative ternary semigroup x = xe2
xTT. It is a contradiction. Hence T has no
idempotent elements. Therefore T is a noetherian, cancellative without idempotents.
(3) (4) : Suppose that T is a noetherian cancellative ternary semigroup without idempotents. Let a T. If
possible, suppose that < a >w
= . Then there exists b T such that b< a >w
. Now b
1i
< a >n
, implies
that b < a >n
for all odd natural numbers n and hence b = ai siti for some si ,ti T for all i = 1, 3, 5, . . . . .
Therefore b = ai siti = ai+2si+1 ti+1 for i = 1, 3, 5, . . . . Now consider yi = ai siti for all i = 1, 3, 5, . . . . .,
then yi = a2 yi+2 for all i = 1, 3, 5, . . . . Therefore < yi > < yi+2 > for each i = 1, 3, 5, . . . . . . Since T is
Noetherian, the chain < y
1
> < y
3
> < y
5
> . . . . . is stationary. Therefore there exists a odd
natural number n such that < yn > = < yn+2 > = < yn+ 5 > = . . . Thus yn+2 = st yn for some s, t T. Now
a2
yn+2 = yn , implies that a2
stsn = sn and hence (a 3st) sn = (a) sn, implies that (a 3st)snsn = a snsn . By
cancellative law, we have a3s = a. Thus a is left regular clement in T. Since T is commutative, a is regular
element in T. Therefore T has regular elements. It is a contradiction. Therefore < a >w
= ..
(4) (5) : Suppose that < a >w
= . for every a T. Since x S, clearly < x >w
= .
10. Chained Commutative Ternary Semigroups
www.iosrjournals.org 58 | Page
(5)(1) : Suppose that < a >w
= . Let a T. If possible, suppose that a xn for any odd natural numbers n.
Therefore a = xs1t1 , s1 , t1 S and s1 xp
, t1 x q
for any odd natural numbers p, q. Similarly s1 = xs2 sl
2
,and t1 = xt2tl
2 where s2 , sl
2, t2 , tl
2 S and each of them not eaqual to xp
for any odd natural number. Hence
a = xs1t1 = x3
s3t3. By continuing this process, we get a = xs1t1 = x3
s3t3 = x5s5 t5 =. . . , therefore a <
xn
>=< x >n
for all odd natural numbers n and hence a
1i
< x >n
= < x >w
= . It is a contradiction.
Therefore a = xn for some odd natural numbers n. Hence S={x, x 3, x3, . . . .}. If T is finite then T = {x, x 3,
x5,. . .., xm} for some odd natural numbers m.
Now < x m
> < x m-2
> . . . . < x > and hence < x >m
< x > m-2
< x > m-4
. . . . < x >.
Also < x > m+r
= < x > m
for all odd natural numbers r. So xm
1i
< x >n
= < x >w
= . It is a contradiction.
Therefore T is infinite.
COROLLARY 3.46 : If T is a commutative chained ternary semigroup with T ≠ T3 , then the following
are equivalent.
(1) S={x, x
3
, x
5
, . . . . . . .}, where x TT3
(2) T is Noetherian cancellative ternary semigroup without regular elements.
(3) < a >w
= for all a T.
Proof : The proof of the theorem follows the above theorem.
THEOREM 3.47: If T is a commutative chained noetherian cancellative ternary semigroup without
regular elements, then < a >w
= for all a T.
Proof : Suppose if possible, T has no proper ideals. Since T is commutative, T has neither proper left/ right/
lateral ideals. Therefore by theorem 3.38, T is a ternary group. By theorem 3.41, every element of T is regular.
It is contradiction to T has no regular elements. Hence T has proper ideals. Since T is noetherian ternary
semigroup and T has proper ideals implies that T contains maximal ideals. Suppose if possible T = T 3
. Since
T contains maximal ideals and T = T 3
, implies by theorem 2.61, T contains regular elements. It is
contradiction. Thus T T 3 , by theorem 3.10, < a >w
= for all a T.
Acknowledgement : The author would like to thank our college vice-president Dr. S.R.K.Prasad for
encouraging me to do this research work.
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