In this paper, we study the class of Right regular and Multiplicatively subidempotent semirings.
Especially we have focused on the additive identity ‘e’ which is also multiplicative identity in both semirings.
The document lists the group members working on limits of complex functions. It then provides two definitions of limits - the first conceptual definition, and the second more precise "epsilon-delta" definition introduced by Bernard Bolzano in 1817. It explains that the epsilon refers to how close the function value needs to be to the limit, while delta refers to how close the input needs to be to still satisfy the epsilon condition. Some examples are then worked out to demonstrate the limit concept. Applications of limits in various engineering and science fields are discussed, such as fluid mechanics, mechanical design, astronomy, business, and medicine. It concludes by noting that what qualifies as "close enough" can depend on the context and precision needed.
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
This document provides information about number theory, including divisors, prime factorization, and congruences. It begins by defining divisors and the division algorithm, and proves several theorems about greatest common divisors and expressing them as linear combinations. It then discusses prime numbers and Euclid's lemma, and proves the fundamental theorem of arithmetic that every integer can be uniquely expressed as a product of prime factors. The document concludes by defining congruences modulo m and listing some basic properties of congruences.
1. The document contains exercises on limit theorems from an introduction to real analysis course. It includes 4 problems asking the student to determine if sequences converge or diverge based on given formulas, provide examples of sequences whose sum and product converge but the individual sequences diverge, and prove statements about convergent sequences.
2. The solutions show work for determining convergence or divergence of sequences defined by formulas in problem 1. Examples are given in problem 2 where the sum and product of divergent sequences converge. Theorems are applied in problems 3 and 4 to prove relationships between convergent sequences.
1) The document uses mathematical induction to prove several formulas.
2) It demonstrates proofs for formulas like 1 + 3 + 5 + ... + (2n-1) = n^2 and 2 + 4 + ... + 2n = n(n+1).
3) The proofs follow the standard structure of mathematical induction, showing the base case is true and using the induction hypothesis to show if the statement is true for n it is also true for n+1.
This document defines and provides examples of metric spaces. It begins by introducing metrics as distance functions that satisfy certain properties like non-negativity and the triangle inequality. Examples of metric spaces given include the real numbers under the usual distance, the complex numbers, and the plane under various distance metrics like the Euclidean, taxi cab, and maximum metrics. It is noted that some functions like the minimum function are not valid metrics as they fail to satisfy all the required properties.
The document lists the group members working on limits of complex functions. It then provides two definitions of limits - the first conceptual definition, and the second more precise "epsilon-delta" definition introduced by Bernard Bolzano in 1817. It explains that the epsilon refers to how close the function value needs to be to the limit, while delta refers to how close the input needs to be to still satisfy the epsilon condition. Some examples are then worked out to demonstrate the limit concept. Applications of limits in various engineering and science fields are discussed, such as fluid mechanics, mechanical design, astronomy, business, and medicine. It concludes by noting that what qualifies as "close enough" can depend on the context and precision needed.
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
This document provides information about number theory, including divisors, prime factorization, and congruences. It begins by defining divisors and the division algorithm, and proves several theorems about greatest common divisors and expressing them as linear combinations. It then discusses prime numbers and Euclid's lemma, and proves the fundamental theorem of arithmetic that every integer can be uniquely expressed as a product of prime factors. The document concludes by defining congruences modulo m and listing some basic properties of congruences.
1. The document contains exercises on limit theorems from an introduction to real analysis course. It includes 4 problems asking the student to determine if sequences converge or diverge based on given formulas, provide examples of sequences whose sum and product converge but the individual sequences diverge, and prove statements about convergent sequences.
2. The solutions show work for determining convergence or divergence of sequences defined by formulas in problem 1. Examples are given in problem 2 where the sum and product of divergent sequences converge. Theorems are applied in problems 3 and 4 to prove relationships between convergent sequences.
1) The document uses mathematical induction to prove several formulas.
2) It demonstrates proofs for formulas like 1 + 3 + 5 + ... + (2n-1) = n^2 and 2 + 4 + ... + 2n = n(n+1).
3) The proofs follow the standard structure of mathematical induction, showing the base case is true and using the induction hypothesis to show if the statement is true for n it is also true for n+1.
This document defines and provides examples of metric spaces. It begins by introducing metrics as distance functions that satisfy certain properties like non-negativity and the triangle inequality. Examples of metric spaces given include the real numbers under the usual distance, the complex numbers, and the plane under various distance metrics like the Euclidean, taxi cab, and maximum metrics. It is noted that some functions like the minimum function are not valid metrics as they fail to satisfy all the required properties.
This document is a dissertation submitted by Amit Kumar Singh for his M.Sc. in Mathematics at the University of Allahabad. It discusses various Diophantine equations including linear equations of the form ax+by=c, Pythagorean triples satisfying x^2 + y^2 = z^2, and Fermat's Last Theorem that x^n + y^n cannot equal z^n for integers when n is greater than 2. The document contains acknowledgments, contents, and sections on the life of Diophantus and different types of Diophantine equations.
1. Set theory is an important mathematical concept and tool that is used in many areas including programming, real-world applications, and computer science problems.
2. The document introduces some basic concepts of set theory including sets, members, operations on sets like union and intersection, and relationships between sets like subsets and complements.
3. Infinite sets are discussed as well as different types of infinite sets including countably infinite and uncountably infinite sets. Special sets like the empty set and power sets are also covered.
This document provides an introduction to set theory. It defines what a set is and provides examples of common sets. A set can be represented in roster form by listing its elements within curly brackets or in set builder form using a characteristic property. There are different types of sets such as finite sets, infinite sets, empty sets, singleton sets, and power sets which contain all subsets. Set operations like union, intersection, difference and symmetric difference are introduced. Important concepts like subsets, equivalent sets, disjoint sets and complements are also covered.
This document provides lecture notes on complex analysis covering four units of content:
1) The index of a close curve, Cauchy's theorem, and entire functions.
2) Counting zeroes, meromorphic functions, and maximum principle.
3) Spaces of continuous and analytic functions, and behavior of functions.
4) Comparison of entire functions, analytic continuation, and harmonic functions.
It also provides definitions and theorems regarding integrals over rectifiable curves, winding numbers, and Cauchy's theorem. Exercises and proofs are included.
The document discusses lines and planes in mathematics. It provides multiple ways to specify a line, including using two points, a point and slope, or a slope and y-intercept. Lines can also be described using vectors, with a line being the set of points a + tv, where a is a point on the line, v is a direction vector, and t is a real number. Planes are similarly defined as the set of points where the dot product of a normal vector p and the offset (x - a) is 0, where a is a point on the plane. An example shows how to check if three points lie on the same line by finding the line equation and checking if a third point satisfies it.
Polynomials are algebraic expressions with multiple terms. They can contain variables, constants, and exponents of 0 or positive integers. Polynomials are easy to work with because adding and multiplying polynomials always results in another polynomial. The degree of a polynomial refers to the highest exponent of its variable terms. Polynomials are written in standard form by ordering terms from highest to lowest degree. Like terms can be combined by adding their coefficients.
This document provides an introduction to number theory, including:
- Number theory is the study of integers and their properties
- It discusses the origins and early developments of number theory in places like Mesopotamia, India, Greece, and Alexandria
- It defines different types of numbers like natural numbers, integers, rational numbers, irrational numbers, and describes properties like prime and composite numbers
- It discusses applications of number theory like public key cryptography and error-correcting codes
Here we focuses on Fixed-Point Iterative Technique for solving nonlinear Equations in Numerical Analysis. It is one of the opened-iterative techniques for finding roots called Fixed-Point of Non-linear Equations.
This document provides an overview of abstract algebra and key concepts such as groups. It discusses how the word "algebra" is derived from an Arabian word meaning "union of broken parts." It also outlines important mathematicians who contributed to the development of algebra, such as Al Khwarizmi, the "father of algebra." The document defines what a set and group are, including the properties a group must satisfy like closure, associative, identity, and inverse elements. Examples of groups are given like integers, rational numbers, and matrices. Applications of group theory in fields like physics, chemistry, and technology are mentioned.
Basic concepts of integration, definite and indefinite integrals,properties of definite integral, problem based on properties,method of integration, substitution, partial fraction, rational , irrational function integration, integration by parts, reduction formula, improper integral, convergent and divergent of integration
This document contains a 20 question matrices worksheet for Class 12 students. It covers topics like addition and multiplication of matrices, inverse of matrices, determinant, rank of matrices, and solving systems of linear equations using matrices. The worksheet is from Sthitpragya Science Classes, an institute providing advanced mathematics coaching for engineering entrance exams like JEE and BITSAT in Gandhidham, Gujarat.
This document discusses sequences and their limits. Some key points:
- A sequence is a list of numbers written in a definite order. It can be thought of as a function with domain the positive integers.
- The limit of a sequence is defined as the number L such that the terms of the sequence can be made arbitrarily close to L by choosing a sufficiently large term.
- A sequence converges if it has a finite limit, and diverges if its terms approach infinity. Bounded monotonic sequences are guaranteed to converge.
- Properties of sequence limits parallel those of limits of functions, including laws of limits and the ability to pass limits inside continuous functions.
1) Mathematical induction is a method of proof that can be used to prove statements for all positive integers. It involves showing that a statement is true for n=1, and assuming it is true for an integer k to prove it is true for k+1.
2) The document provides an example using mathematical induction to prove the formula Sn = n(n+1) for the sum of the first n even integers.
3) Finite differences are used to determine if a sequence has a quadratic model by seeing if the second differences are constant. The example finds the quadratic model n^2 for the sequence 1, 4, 9, 16, 25, 36.
In this presentation you will able to know about permutation group or symmetric group. You will also learn about degree and order of permutation group. The permutation group is containing only even permutations is called Alternating group.
This document discusses permutations and combinations. Permutations refer to the number of arrangements that can be made by selecting some or all items from a set. Combinations refer to the number of groups that can be formed by selecting some items from a set. The document provides formulas for calculating permutations and combinations, and examples of applying these concepts such as calculating the number of ways to form a cricket team from available players.
Graph isomorphism is the problem of determining whether two graphs are topologically identical. It is in NP but not known to be NP-complete. The refinement heuristic iteratively partitions vertices into equivalence classes to reduce possible mappings between graphs. For trees, there is a linear-time algorithm that labels vertices based on subtree structure to test rooted and unrooted tree isomorphism.
This document introduces the topic of graph theory. It defines what graphs are, including vertices, edges, directed and undirected graphs. It provides examples of graphs like social networks, transportation maps, and more. It covers basic graph terminology such as degree, regular graphs, subgraphs, walks, paths and cycles. It also discusses graph classes like trees, complete graphs and bipartite graphs. Finally, it touches on some historical graph problems, complexity analysis, centrality analysis, facility location problems and applications of graph theory.
Teks tersebut merangkum perkembangan geometri non-Euclid, dimulai dari kontribusi para matematikawan Arab dan Eropa dalam mempertanyakan postulat kelima Euclid, hingga penemuan geometri hiperbolik dan non-Euclid oleh Gauss, Lobachevsky, dan Bolyai pada abad ke-19. Saccheri dianggap sebagai pendiri geometri non-Euclid karena membuktikan bahwa jumlah sudut segitiga kurang dari 180 derajat.
1) The document contains exercises on the completeness property of real numbers from an introduction to real analysis course.
2) It includes problems, solutions, and proofs regarding the infimum and supremum of sets, bounded sets, and the completeness property.
3) Key results proven include if A and B are bounded subsets of the real numbers, then their union A ∪ B is bounded, and if S is a bounded set and S0 is a nonempty subset of S, then the infimum of S0 is greater than or equal to the infimum of S.
This document presents definitions and properties related to left regular semirings. It begins by defining key concepts such as semirings, left singular semigroups, and left regular semirings. It then establishes several theorems about the structures of left regular semirings, including: if a left regular semiring has a multiplicative identity that is also an additive identity, then its multiplicative reduct is quasi separative; if its additive reduct is a zeroid, then it is a Viterbi semiring; and if its additive reduct is positively totally ordered, then addition is the same as multiplication. The document concludes by providing an example to illustrate that a semiring is left regular if its additive and multiplic
This document discusses properties of LA-semirings where the multiplicative structure (S,.) is an anti-inverse semigroup. It proves several theorems about the structures of the additive and multiplicative semigroups in such LA-semirings. Specifically, it shows that if a LA-semiring satisfies the identity a+1=1, then (1) the additive semigroup (S,+) is anti-inverse and abelian, and (2) the sum of two anti-inverse elements is again anti-inverse. It also proves that if the multiplicative semigroup is anti-inverse, then the additive semigroup and product of elements are abelian.
This document is a dissertation submitted by Amit Kumar Singh for his M.Sc. in Mathematics at the University of Allahabad. It discusses various Diophantine equations including linear equations of the form ax+by=c, Pythagorean triples satisfying x^2 + y^2 = z^2, and Fermat's Last Theorem that x^n + y^n cannot equal z^n for integers when n is greater than 2. The document contains acknowledgments, contents, and sections on the life of Diophantus and different types of Diophantine equations.
1. Set theory is an important mathematical concept and tool that is used in many areas including programming, real-world applications, and computer science problems.
2. The document introduces some basic concepts of set theory including sets, members, operations on sets like union and intersection, and relationships between sets like subsets and complements.
3. Infinite sets are discussed as well as different types of infinite sets including countably infinite and uncountably infinite sets. Special sets like the empty set and power sets are also covered.
This document provides an introduction to set theory. It defines what a set is and provides examples of common sets. A set can be represented in roster form by listing its elements within curly brackets or in set builder form using a characteristic property. There are different types of sets such as finite sets, infinite sets, empty sets, singleton sets, and power sets which contain all subsets. Set operations like union, intersection, difference and symmetric difference are introduced. Important concepts like subsets, equivalent sets, disjoint sets and complements are also covered.
This document provides lecture notes on complex analysis covering four units of content:
1) The index of a close curve, Cauchy's theorem, and entire functions.
2) Counting zeroes, meromorphic functions, and maximum principle.
3) Spaces of continuous and analytic functions, and behavior of functions.
4) Comparison of entire functions, analytic continuation, and harmonic functions.
It also provides definitions and theorems regarding integrals over rectifiable curves, winding numbers, and Cauchy's theorem. Exercises and proofs are included.
The document discusses lines and planes in mathematics. It provides multiple ways to specify a line, including using two points, a point and slope, or a slope and y-intercept. Lines can also be described using vectors, with a line being the set of points a + tv, where a is a point on the line, v is a direction vector, and t is a real number. Planes are similarly defined as the set of points where the dot product of a normal vector p and the offset (x - a) is 0, where a is a point on the plane. An example shows how to check if three points lie on the same line by finding the line equation and checking if a third point satisfies it.
Polynomials are algebraic expressions with multiple terms. They can contain variables, constants, and exponents of 0 or positive integers. Polynomials are easy to work with because adding and multiplying polynomials always results in another polynomial. The degree of a polynomial refers to the highest exponent of its variable terms. Polynomials are written in standard form by ordering terms from highest to lowest degree. Like terms can be combined by adding their coefficients.
This document provides an introduction to number theory, including:
- Number theory is the study of integers and their properties
- It discusses the origins and early developments of number theory in places like Mesopotamia, India, Greece, and Alexandria
- It defines different types of numbers like natural numbers, integers, rational numbers, irrational numbers, and describes properties like prime and composite numbers
- It discusses applications of number theory like public key cryptography and error-correcting codes
Here we focuses on Fixed-Point Iterative Technique for solving nonlinear Equations in Numerical Analysis. It is one of the opened-iterative techniques for finding roots called Fixed-Point of Non-linear Equations.
This document provides an overview of abstract algebra and key concepts such as groups. It discusses how the word "algebra" is derived from an Arabian word meaning "union of broken parts." It also outlines important mathematicians who contributed to the development of algebra, such as Al Khwarizmi, the "father of algebra." The document defines what a set and group are, including the properties a group must satisfy like closure, associative, identity, and inverse elements. Examples of groups are given like integers, rational numbers, and matrices. Applications of group theory in fields like physics, chemistry, and technology are mentioned.
Basic concepts of integration, definite and indefinite integrals,properties of definite integral, problem based on properties,method of integration, substitution, partial fraction, rational , irrational function integration, integration by parts, reduction formula, improper integral, convergent and divergent of integration
This document contains a 20 question matrices worksheet for Class 12 students. It covers topics like addition and multiplication of matrices, inverse of matrices, determinant, rank of matrices, and solving systems of linear equations using matrices. The worksheet is from Sthitpragya Science Classes, an institute providing advanced mathematics coaching for engineering entrance exams like JEE and BITSAT in Gandhidham, Gujarat.
This document discusses sequences and their limits. Some key points:
- A sequence is a list of numbers written in a definite order. It can be thought of as a function with domain the positive integers.
- The limit of a sequence is defined as the number L such that the terms of the sequence can be made arbitrarily close to L by choosing a sufficiently large term.
- A sequence converges if it has a finite limit, and diverges if its terms approach infinity. Bounded monotonic sequences are guaranteed to converge.
- Properties of sequence limits parallel those of limits of functions, including laws of limits and the ability to pass limits inside continuous functions.
1) Mathematical induction is a method of proof that can be used to prove statements for all positive integers. It involves showing that a statement is true for n=1, and assuming it is true for an integer k to prove it is true for k+1.
2) The document provides an example using mathematical induction to prove the formula Sn = n(n+1) for the sum of the first n even integers.
3) Finite differences are used to determine if a sequence has a quadratic model by seeing if the second differences are constant. The example finds the quadratic model n^2 for the sequence 1, 4, 9, 16, 25, 36.
In this presentation you will able to know about permutation group or symmetric group. You will also learn about degree and order of permutation group. The permutation group is containing only even permutations is called Alternating group.
This document discusses permutations and combinations. Permutations refer to the number of arrangements that can be made by selecting some or all items from a set. Combinations refer to the number of groups that can be formed by selecting some items from a set. The document provides formulas for calculating permutations and combinations, and examples of applying these concepts such as calculating the number of ways to form a cricket team from available players.
Graph isomorphism is the problem of determining whether two graphs are topologically identical. It is in NP but not known to be NP-complete. The refinement heuristic iteratively partitions vertices into equivalence classes to reduce possible mappings between graphs. For trees, there is a linear-time algorithm that labels vertices based on subtree structure to test rooted and unrooted tree isomorphism.
This document introduces the topic of graph theory. It defines what graphs are, including vertices, edges, directed and undirected graphs. It provides examples of graphs like social networks, transportation maps, and more. It covers basic graph terminology such as degree, regular graphs, subgraphs, walks, paths and cycles. It also discusses graph classes like trees, complete graphs and bipartite graphs. Finally, it touches on some historical graph problems, complexity analysis, centrality analysis, facility location problems and applications of graph theory.
Teks tersebut merangkum perkembangan geometri non-Euclid, dimulai dari kontribusi para matematikawan Arab dan Eropa dalam mempertanyakan postulat kelima Euclid, hingga penemuan geometri hiperbolik dan non-Euclid oleh Gauss, Lobachevsky, dan Bolyai pada abad ke-19. Saccheri dianggap sebagai pendiri geometri non-Euclid karena membuktikan bahwa jumlah sudut segitiga kurang dari 180 derajat.
1) The document contains exercises on the completeness property of real numbers from an introduction to real analysis course.
2) It includes problems, solutions, and proofs regarding the infimum and supremum of sets, bounded sets, and the completeness property.
3) Key results proven include if A and B are bounded subsets of the real numbers, then their union A ∪ B is bounded, and if S is a bounded set and S0 is a nonempty subset of S, then the infimum of S0 is greater than or equal to the infimum of S.
This document presents definitions and properties related to left regular semirings. It begins by defining key concepts such as semirings, left singular semigroups, and left regular semirings. It then establishes several theorems about the structures of left regular semirings, including: if a left regular semiring has a multiplicative identity that is also an additive identity, then its multiplicative reduct is quasi separative; if its additive reduct is a zeroid, then it is a Viterbi semiring; and if its additive reduct is positively totally ordered, then addition is the same as multiplication. The document concludes by providing an example to illustrate that a semiring is left regular if its additive and multiplic
This document discusses properties of LA-semirings where the multiplicative structure (S,.) is an anti-inverse semigroup. It proves several theorems about the structures of the additive and multiplicative semigroups in such LA-semirings. Specifically, it shows that if a LA-semiring satisfies the identity a+1=1, then (1) the additive semigroup (S,+) is anti-inverse and abelian, and (2) the sum of two anti-inverse elements is again anti-inverse. It also proves that if the multiplicative semigroup is anti-inverse, then the additive semigroup and product of elements are abelian.
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
A weaker version of continuity and a common fixed point theoremAlexander Decker
This article presents a generalization of previous theorems on common fixed points of self-maps. It introduces the concept of property E.A. and weak compatibility between self-maps. A new theorem (Theorem B) is proved which finds a unique common fixed point for three self-maps under weaker conditions than previous results, including relaxing orbital completeness and removing the requirement of orbital continuity. The proof of Theorem B is provided. It is shown that this new theorem generalizes an earlier result from the literature.
This document provides problems and solutions related to mathematics. It contains proposed problems submitted for consideration to be published, as well as solutions submitted in response to previously published problems. The problems cover a range of mathematical topics and include proving identities, inequalities, and other relationships. The solutions thoroughly explain the reasoning and steps to solve each problem.
This document contains a sample question paper for Class XII Mathematics. It has 5 sections (A-E). Section A contains 18 multiple choice questions and 2 assertion-reason questions worth 1 mark each. Section B has 5 very short answer questions worth 2 marks each. Section C contains 6 short answer questions worth 3 marks each. Section D has 4 long answer questions worth 5 marks each. Section E contains 3 case study/passage based questions worth 4 marks each with internal subparts. The document provides sample questions on topics including trigonometry, calculus, matrices, probability, linear programming and more.
This document outlines the course contents, schedule, and evaluation for CSE 173: Discrete Mathematics taught by Dr. Saifuddin Md.Tareeq at DU. The course covers topics like logic, sets, functions, algorithms, number theory, induction, counting, probability, relations, and graphs. It will be evaluated based on homework, quizzes, midterms, and a final exam. Discrete mathematics is the study of discrete rather than continuous structures, and concepts from it are useful for computer algorithms, programming, cryptography, and software development.
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces.
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the
conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
This document discusses subspaces, spanning sets, and bases in vector spaces. Some key points:
- A subspace is a subset of a vector space that is closed under vector addition and scalar multiplication.
- The span of a set S in a vector space V is the smallest subspace of V containing S.
- A set S is a basis for V if every vector in V can be uniquely written as a linear combination of vectors in S.
- Examples are provided to illustrate subspaces, spans, and finding the coordinate representation of vectors with respect to given bases.
The document discusses alpha cuts and their properties in fuzzy sets. It defines alpha cuts as crisp sets containing elements of the universal set whose membership degree in the fuzzy set is greater than or equal to alpha. The higher the alpha value, the smaller the alpha cut set. It also discusses support, core, and height of fuzzy sets. Support is the crisp set of all elements with non-zero membership, core those with membership 1, and height the highest alpha value of a non-empty alpha cut. Examples are given to illustrate key fuzzy set operations and concepts.
This document contains solutions to homework problems involving set theory concepts like unions, intersections, complements, Cartesian products, and Venn diagrams. Key ideas summarized include determining the members of specific sets defined using set builder notation, evaluating statements about subset and equality relationships between sets, using Venn diagrams to illustrate set relationships, finding cardinalities of finite sets, and expressing set operations in terms of logic operators and simplifying using set identities.
This document contains information from a class presentation on computing elements of one-dimensional arrays and recursively defined sequences. It discusses how to count the elements of an array that is cut in the middle, the probability of an element having an even or odd subscript, computing terms of a recursively defined sequence, and properties of relations. The presentation was given by 5 students to their instructor for their BS in Computer Science class from 2013-2017.
The document discusses the differences and relationships between quadratic functions and quadratic equations. It notes that quadratic functions can take any real number as an input, while quadratic equations only have two solutions. The roots of a quadratic equation are also the x-intercepts of the graph of the corresponding quadratic function. The remainder theorem states that the value of a polynomial when a number is substituted for the variable is equal to the remainder when the polynomial is divided by the linear factor corresponding to that number. This connects the roots of quadratic equations to factors of quadratic functions. A quadratic can only have two distinct roots, as having three would mean it has an infinite number of roots.
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
This document provides an overview of unit 2 on the algebra of vectors from the course EMA 310: Vectors and Mechanics. It introduces the learning objectives which are to find the resultant of given vectors, add vectors using the parallelogram and triangle laws of addition, and establish and use properties of vector addition. Examples are given of applying the triangle and parallelogram laws of vector addition, along with activities for students to practice finding vector sums and multiplying vectors by scalars.
This document presents a dissertation on module theory submitted in partial fulfillment of a master's degree. It contains an introduction, three chapters, and a conclusion. Chapter 1 provides preliminaries on groups, rings, vector spaces, and related concepts needed to understand modules. Chapter 2 introduces modules and submodules, discusses module homomorphisms, quotient modules, generation of modules, and direct sums. Chapter 3 examines Artinian and Noetherian modules, which have special properties regarding ascending and descending chains of submodules.
This document summarizes numerical simulations conducted using the VecTor2 software to model the bond stress-slip effect of reinforced concrete on the pushover behavior of walls. The simulations considered the influence of bond stress-slip on the shear force, crack patterns, and displacement of reinforced concrete walls under loading. The results of the VecTor2 simulations were compared to experimental test data. The simulations were able to predict the lateral force, crack patterns, and displacements observed in the experimental tests. The simulations also provided stress and strain distributions in the walls. The bond stress-slip models in VecTor2 helped predict the wall behavior observed in experiments.
The concept of sustainable construction is increasingly affecting the development of the construction market.The specificity of construction as an economic activity and ofthe construction product (goods and services) determine the existence of a complex vertical chain of links, involving different actors, who tend to work in the short term and are limited to the rational use of knowledge and experience in practice. Moreover, it is characterized by a low level of inter-company relationshipsresulting in a fragmentation of responsibilityand complicates and hinders the realization of projects and sites,which meet the requirements of sustainable construction. Sustainable construction requires a holistic approach and substantial changes in the organization of construction activity, both at the market and firm level, under the active role of the state. The aim of the study is: 1) analysis of problems in the vertical chain of connections in the construction market, 2)an analysis of the possibilities for creating stable long-term relationships and a joint approach of clients, contractors and subcontractors, which can provide economic, social and environmental efficiency of the construction.
Since the recent development of UAVs(Unmanned Aerial Vehicles) and digital sensors technology has enabled the acquisition of high-resolution image data, it is considered that the image data of riverside can be analysed. Therefore, this study analyses the applicability of remote sensing techniques through image analysis in river systems and habitats. The target stream in this study was the Cheongmi stream and the applicability of the river environmental evaluation technique was analysed through image analysis. The satellite images used for the analysis of river topography and environments were compared with the aerial images taken by a micro UAV), and the river environmental evaluation was carried out with the field research at the same time. The data acquisition range and application limit by river environmental evaluation technique proposed previously were evaluated, and as a result, it was found that it was possible to draw various evaluation parameters using a drone that could take an image at a low altitude in comparison to satellite images.
Industrial engineering is founded on the idea that there is always a better way. This mantra rings true in everything an industrial engineer does, from lean manufacturing to six sigma, to quality control and ergonomics. This paper demonstrates the uniqueness of this discipline, the impact its techniques has in sectors outside of manufacturing, and the positive effects it has on businesses.
The study was carried out using the UAV for analyzing the characteristics of debris in order to present the methodology to estimate the quantitative amount of debris caught in small river facilities. A total of six small rivers that maintained the form of a natural river were selected for collecting UAV images, and the grouping of each target in the image was carried out using the object-based classification method, and based on the object-based classification result of the UAV images, the land cover classification for the status of factors causing the generation of debris for six target sections was carried out by applying the screen digitizing method. In addition, in order to verify the accuracy of the classification result, the error matrix was performed, securing the reliability of the result. The accuracy analysis result showed that for all six target sections, the overall accuracy was 93.95% and the Kappa coefficient was 0.93, showing an excellent result.
Multilevel Inverters are getting popular and have become more attractive to researchers in the recent times for high power applications due to their better power quality and higher efficiency as compared to two level inverters. This research work presents a detailed comparative analysis of various multicarrier sinusoidal PWM schemes such as In Phase Disposition, Phase Opposition Disposition and Alternate Phase Opposite Disposition implemented on five level conventional and modified cascaded h-bridge inverters in MATLAB/SIMULINK software. Conventional five level topology uses eight switches and suffers from increased switching complexity while modified five level topology uses only five switches and is recommended to reduce switching complexity and switching losses. It also ensures less number of components, reduced size and overall cost of the system. The effect of modulation index (Ma) on the output harmonic contents in various PWM techniques is also analyzed.
Objective: Cervical cancer (CC) is one of the leading causes of cancer-related deaths among women worldwide.Human papillomavirus (HPV) is the most important element in this disease.The aim of this study is to prepare TiO2/ZnO nanocomposite (NC), titanium dioxide (TiO2) and zinc oxide (ZnO)nanoparticles (NPs) to determine the anticancer activity on human CC cell line (HeLa) and healthy mouse fibroblast cell line (L-929). Materials&Methods: ZnO, TiO2 NPs and NC were prepared by a solution combustion synthesis method. The samples were characterized by ultraviolet–visible spectroscopy. Stability analysis was performed with zeta potential. The synthesized NC and NPs were permormed to the HeLa and L-929 cell lines and anticancer activity of these NC and NPs were determined by using MTT method. The HeLa and L-929 cells were treated with different concentrations of these NC and NPs (0,5-100 μg/ml) for 24, 48 and 72 hours. The spectrophotometric readings at 570 nm were recorded and analysed with Graphpad Prism7. Results: NC and NPs were successfully synthesized. The effects of these NC and NPs on the HeLa and L-929 cells were compared with the control group and IC50 values were determined for 24, 48 and 72 hours. Then we compared the effects of these molecules on the L-929 cell line with the HeLa cell line and founded more active is on HeLa cells. Conclusion:There are many drugs used in CC treatment. However, undesirable toxicity and drug resistance of these drugs negatively affect treatment.We have synthesized NC and NPs in order to formulate basis of a new drug in this study and have identified anti-cancer activity.As a result, we found that NC and NPs anti-cancer activity was higher in HeLa cells than in L-929.
Graphene is a material that attracts attention in technical textile applications as in many other areas due to its outstanding features. In this study, it was aimed to investigate the performance properties of graphene coated fabrics. Pre-treated polyester fabrics were coated with nano-graphene powders at different concentration rates (50, 100 and 200 g/kg) by knife-over-roll technique. According to test results, generally, the graphene coating had a positive effect on the performance properties of polyester fabrics.
This study was focused on the effects of Sugarcane Bagasse Ash (SCBA) additive on process parameters and compost quality of Co-composting of filter cake and bagasse. Filter cake and bagasse were mixed and sugar cane bagasse ash (SCBA) from a heating power plant of sugar mill. Three compost mixes (M) were obtained: MA with 0%, MB with 10% and MC with 20 wt % of fuel ash. These three different mixes were composted in an experimental composter as three parallel experiments for 3 weeks each. The physical, chemical and biological parameters were monitoring during composting. Significantly, ash additives decreased the total organic carbon; measured by mineralization the breaking down of the organic matter was more rapid in the MC than in the MA, as well as increased the pH during composting. Interesting, the pH decreased was most important in MA and attend 5 for the first week of composting, and then it gradually increased to pH around 8 at the end of the process. The results indicated that ash inhibits the pH drop due to production of organic acids during composting. The acidity of the material was reported as affects the process during the initial phase of rising temperature and quality of the final product. The temperature reached up to 50-55oC during thermophilic phase, the greater temperature was obtained for MC. At the end of composting, the electrical conductivity increased in the MC, especially in MC, but don’t exceed limit (4 mS/cm) for prevent phytotoxicity of the compost. The SCBA additive was likely to speed up the composting process of bagasse with filter cake from 44 days to 33 days.
The work presents report on production and analysis of bioresin from epoxidized mango kernel oil (EMKO). The bioresin (acrylated epoxidized mango kernel oil) or AEMKO was produced from epoxidized mango kernel oil via acrylation chemical reaction route. The FTIR spectrum analysis of epoxidized mango kernel oil (EMKO) and acrylated epoxidized mango kernel oil (AEMKO) produced gave the degree of acrylation (DOA) as 46%. The Viscosity of AEMKO (resin) was determined at room temperature (25 °C) to be 387cP while the density at 25oC was 1.2 g/cm3. The glass transition temperature (Tg) of the bioresin was determined to be 95oC. Production cost analysis of the bioresin was done and found to be N8, 804.35 per litre. The high cost was due to high costs of the chemicals, labour and overhead charges involved at my local level. At commercial level, those components of the costs would definitely reduce to the level compatible with synthetic (polyester) resin (N2, 500 per litre) currently sold by some markers in Nigeria. However, the overall results of the work demonstrated that bioresin can be successfully synthesized from mango kernel oil with properties compatible with ASTM standards. The commercial production of the bioresin will go a long way in mitigating some of the challenges associated with total use of fossil fuel currently use for production of bulk of synthetic resins for composite manufacturing activities.
The window functions used for digital filter design are used to eliminate oscillations in
the FIR (Finite Impulse Response) filter design. In this work, the use of Particle Swarm Optimization
(PSO) algorithm is proposed in the design of cosh window function, in which has widely used in the
literature and has useful spectral parameters. The cosh window is a window function derived from the
Kaiser window. It is more advantageous than the Kaiser window because there is no power series
expansion in the time domain representation. The designed window function shows better ripple ratio
characteristics than other window functions commonly used in the literature. The results obtained
were presented in tables and figures and successful results were obtained
The aim of the study was to investigate the relationship between 2D gray scale pixels and 3D gray scale pixels of image reconstructions in computed tomography (CT). The 3D space image reconstruction from data projection was a challenging and difficult research problem. The image was normally reconstructed from the 2D data from CT data projection. In this descriptive study, a synthetics 3D Shepp-Logan phantom was used to simulate the actual data projection from a CT scanner. Real-time data projection of a human abdomen was also included in this study. Additionally, the Graphical User Interface (GUI) for the application was designed using Matlab Graphical User Interface Development Environment (GUIDE). The application was able to reconstruct 2D and 3D images in their respective spaces successfully.The image reconstruction for CT in 3D space was analyzedalong with 2D space in order to show their relationships and shared properties for the purpose of constructing these images.
In this work the antimicrobial activity and the economic viability analysis of the essential oil extracted from the hybrid formed by the seeds species of the Murupi (Capsicum chinense), Criollos de Morellos (Capsicum annuum) and Finger of the young (Capsicum baccatum ). The essential oil of the pepper was obtained by the steam drag process and for this extraction, the Soxhlet method was used. For the determination of the antimicrobial activity of the oil the disc diffusion method was used for the strains of Bacillus cereus, Staphylococcus aureus and Escherichia coli. The results point out the resistance of the tested strains to the essential oil of the respective pepper and, in terms of financial and economic aspects, this was not feasible on a small scale. It is suggested that other microorganisms be tested and, later, that studies be carried out with the purpose of characterizing the studied oil chemically for proper application in the agroindustry.
Eliminating Gibbs phenomenon, which occurs during design of Finite Impulse Response (FIR) digital filter and which is undesirable, is very important in order to provide expected performance from digital filter. Window functions have been developed to eliminate these oscillations and to improve the performance of the filter in this regard. In this work, an application was developed for designing window function using LABVIEW which is a graphical programming environment produced by National Instruments. LABVIEW offers a powerful programming environment away from complexity. In this work, the performances of cosh and exponential window functions, which are designed by using the possibilities of LABVIEW in programming, are examined and the situations that will occur under various conditions are compared.
Better efficiency of the air transport system of a country at the national level, especially in terms of its
capacity to generate value for passenger flow and cargo transport, effectively depends on the identification of
the demand generation potential of each hub for this type of service. This requires the mapping of the passenger
flow and volume of cargo transport of each region served by the system and the number of connections. The
main goal of this study was to identify important factors that account for the great variability (demand) of
regional hubsof the airport modal system in operation in the State of São Paulo, the most populated and
industrialized in the Southeast region in Brazil. For this purpose, datasets for each airport related to passengers
or cargo flow were obtained from time series data in the period ranging from January 01, 2008 to December
31, 2014. Different data analysis approaches could imply in better mapping of the flow of the air modal system
from the evaluation of some factors related to operations/volume. Therefore, different statistical models - such
as multiple linear regression with normal errors and new stochastic volatility (SV) models - are introduced in
this study, to provide a better view of the operation system in the four main regional hubs, within a large group
of 32 airports reported in the dataset.
Linear attenuation coefficient (휇) is a measure of the ability of a medium to diffuse and absorb radiation. In the interaction of radiation with matter, the linear absorption coefficient plays an important role because during the passage of radiation through a medium, its absorption depends on the wavelength of the radiation and the thickness and nature of the medium. Experiments to determine linear absorption coefficient for Lead, Copper and Aluminum were carried out in air. The result showed that linear absorption Coefficient for Lead is 0.545cm – 1, Copper is 0.139cm-1 and Aluminum is 0.271cm-1 using gamma-rays. The results agree with standard values.
This study presents results of Activity Concentrations, Absorbed dose rate and the Annual Effective dose rates of naturally occurring radionuclides (40K, 232Th and 226Ra) absorbed in 8 soil samples collected from different areas within the Ajiwei mining sites in Niger State, North Central Nigeria. A laboratory γ-ray spectrometry NaI (Tl) at the Centre for Energy Research and Training (CERT), Ahmadu Bello University Zaria, was used to carry out the analysis of the soil samples. The values of Activity Concentration for 40K ranged from 421.6174 ± 7.9316 to 768.7403 ± 7.9315; for 226Ra it ranged from 20.6257 ± 2.0858 to 44.0324 ± 5.0985 and for 232Th the ranged is from 23.7172 ± 1.3683 to 62.7137 ± 4.1049 Bq.Kg-1. While the Absorbed Dose for 40K ranged from 17.5814 ± 0.3307 to 32.0565 ± 0.3307 ŋGy.h-1, for 226Ra the range is from 9.5291 ± 0.9636 to 20.3430 ± 2.3555 ŋGy.h-1 and for 232Th range from 14.3252 ± 0.4414 to 37.8791 ± 2.4794 ŋGy.h-1. The total average Absorbed Dose rate of the 8 soil samples collected is 63.7877 ŋGy.h-1 and the estimated Annual Effective Dose for the sampled areas range from 0.0636- 0.1028mSvy-1 (i.e 64 – 103 μSv.y-1), with an average Annual Effective Dose of 0.0782 mSv.y-1 (i.e. 78.2 μSv.y-1). These results show’s that the radiation exposure level reaching members of the public in the study areas is lower than the recommended limit value of 1 mSv.y-1 (UNSCEAR, 2000). Also the mean Radium Equivalents obtained ranged from 107.3259 BqKg-1 (AJ1) to 179.4064 BqKg-1 (AJ4). These results show that the recommended Radium Equivalent Concentration is ≤ 370 BqKg-1 which is the requirement for soil materials to be used for dwellings, this implies that the soil from this site is suitable use for residential buildings. The mean External Hazard Index ( Hext ) ranged from 0.1229 Bqkg-1 (AJ3) to 0.4226 Bqkg-1 (AJ7).. While the maximum allowed value of (Hext = 1) corresponds to the upper limit of Raeq (370 BqKg-1) in order to limit the external gamma radiation dose from the soil materials to 1.5 mGy y-1. That is, this Index should be equal to or less than unity (Hext ≤ = 1). Furthermore, the mean Internal Hazard Index (Hext) ranged from 0.3456 Bqkg-1 (AJ1) to 0.6453 Bqkg-1 (AJ2) .Finally, the mean value of the Excess Alpha Radiation (Iα) ranged from 0.1031 Bq.Kg-1 (AJ1) to 0.2202 Bq.Kg-1 (AJ3. All these values for Iα are below the maximum permissible value of Iα= 1 which corresponds to 200 Bq.Kg-1. It can therefore be said that no radiological hazard is envisaged to dwellers of the study areas and the miners working on those sites area.
Pick and place task is one among the most important tasks in industrial field handled by “Selective
Compliance Assembly Robot Arm” (SCARA). Repeatability with high-speed movement in horizontal plane is
remarkable feature of this type of manipulator. The challenge of design SCARA is the difficulty of achieving
stability of high-speed movement with long length of links. Shorter links arm can move more stable. This
condition made the links should be considered restrict then followed by restriction of operation area
(workspace). In this research, authors demonstrated on expanding SCARA robot’s workspace in horizontal area
via linear sliding actuator that embedded to base link of the robot arm. With one additional prismatic joint the
previous robot manipulator with 3 degree of freedom (3-DOF), 2 revolute joints and 1 prismatic joint is become
4-DOF PRRP manipulator. This designation increased workspace of robot from 0.5698m2 performed by the
previous arm (without linear actuator) to 1.1281m2 by the propose arm (with linear actuator). The increasing
rate was about 97.97% of workspace with the same links length. The result of experimentation also indicated
that the operation time spent to reach object position was also reduced.
This document discusses air and moisture permeability of textile layers through numerical simulation methods. It begins by defining air permeability and describing a common method to determine permeability parameters by comparing experimental flow data to an equation. It then provides examples of simulating idealized fabric structures and combined cloth layers to model airflow. The document concludes by discussing a recommended three-layer clothing combination and analyzing moisture transport and insulation properties using physical principles.
Physical and chemical properties of host environment to concrete structures have serious impact on
the performance and durability of constructed concrete facilities. This paper presents a 7-month study that
simulated the influence of soil contamination due to organic abattoir waste and indiscriminate disposal of spent
hydrocarbon on strength and durability of embedded concrete. Concrete mix, 1:1.5:3 was designed for all cube
and beam specimens with water-cement ratio of 0.5 and the compressive and flexural strengths of the specimen
were measured from age 28 days up to 196 days in the host environment. It was found that both host
environments attack the physical and strength of concrete in compression and flexure. However, hydrocarbon
had much greater adverse effect on the load-carrying capacity of concrete structures and hence make
constructed facilities less serviceable and vulnerable to premature failure.
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1. International Journal of Modern Research in Engineering and Technology (IJMRET)
www.ijmret.org Volume 1 Issue 2 ǁ July 2016.
w w w . i j m r e t . o r g Page 5
A Study on the class of Semirings
M.Amala*1, N.Sulochana2 and T.Vasanthi3
1,3Dept. of Applied Mathematics, Yogi Vemana University, Kadapa, Andhra Pradesh, India
2Asst. Prof., K.S.R.M College of Engineering, Kadapa, Andhra Pradesh, India
E‐mail: amalamaduri@gmail.com, sulochananagam@gmail.com and vasanthitm@gmail.com
Abstract : In this paper, we study the class of Right regular and Multiplicatively subidempotent semirings.
Especially we have focused on the additive identity ‘e’ which is also multiplicative identity in both semirings.
Keywords: Idempotent semiring, Multiplicatively subidempotent semiring, Periodic, positively totally ordered,
Rectangular band.
I. INTRODUCTION:
Various concepts of regularity on semigroups have been investigated by R.Croisot. His studies have
been presented in the book of Clifford A.H. and G.B.Preston as R.Croisot theory one of the central places in the
theory is held by the left regularity. K.S.S. Nambooripad studied on the structures of regular semigroups. we
introduce the notion of Right regular semiring as a generalization of regular semiring. Sen, Ghosh &
Mukhopadhyay studied the congruences on inverse semirings with the commutative additive reduct and Maity
improved this to the regular semirings with the set of all additive idempotents a bi semilattice.
The study of regular semigroups has yielded many interesting results. These results have applications
in other branches of algebra and analysis. Section one deals with introduction. Section two contains definitions.
In third section we study on the class of right regular semiring. In last section we have given some results on
ordered multiplicatively subidempotent semiring.
II. PRELIMINARIES:
Definition 2.1:
A semiring S is a Right regular semiring, if S satisfies the identity a + xa + a = a for all a,
x in S.
Definition 2.2:
A semigroup (S, +) is rectangular band if a = a + x + a for all a, x in S.
A semigroup (S, •) is rectangular band if a = axa for all a, x in S.
Definition 2.3:
An element a in a semigroup (S, +) is periodic if ma = na where m and n are positive integers. A
semigroup (S, +) is periodic if every one of its elements is periodic.
Definition 2.4:
A semiring S is said to be idempotent if a + a = a and a2
= a for all a in S.
Definition 2.5:
In a totally ordered semiring (S, +, •, )
(i) (S, +, ) is positively totally ordered (p.t.o), if a + x a, x for all a, x in S
(ii) (S, •, ) is positively totally ordered (p.t.o), if ax a, x for all a, x in S.
III. CLASSES OF RIGHT REGULAR SEMIRING:
In this section, the structures of Right regular semirings with different semiring properties are given.
We have also framed examples in this chapter.
Lemma 3.1: Let S be a semiring which contains the additive and multiplicative identity „e‟. Then a (≠ e) in S is
a Right regular element if and only if xa = a for all x in S.
2. International Journal of Modern Research in Engineering and Technology (IJMRET)
www.ijmret.org Volume 1 Issue 2 ǁ July 2016.
w w w . i j m r e t . o r g Page 6
Proof: By hypothesis e is additive identity also multiplicative identity then a.e = e.a = a and e + a = a + e =
a for all a, e in S
Since a in S is a Right regular element then a + xa + a = a for all x in S which implies [e + x] a + a = a
xa + a = a xa = a
Therefore xa = a for all x in S
Now we have to prove that a € S is a Right regular element
For this assume that xa = a for all x in S
Let us consider a + xa + a
The above equation can be written as [e + x] a + a = xa + a = xa = a
Thus a + xa + a = a for all x in S
Hence a in S is a Right regular element
Theorem 3.2: Let S be a semiring which contains an additive identity „e‟ also multiplicative identity. If a (≠ e)
in S is a Right regular element, then a + x = a for all x in S.
Proof: Given that a in S is a Right regular element then a + xa + a = a
By adding „x‟ on both sides we get a + xa + a + x = a + x
This implies a + xa + x = a + x a + x (a + e) = a + x
a + xa = a + x xa = a + x
Using above lemma xa = a the above equation reduces to the form
a + x = a for all x in S
Proposition 3.3: If S is a Right regular semiring with multiplicative identity „1‟and (S, •) is a rectangular
band, then (S, +) is periodic.
Proof: Since S is Right regular semiring then a + xa + a = a for all a, x in S
This can also be written as a2
+ axa + a2
= a2
Given that (S, •) is rectangular band then axa = a for all a, x in S
a.1.a + axa + a.1.a = a.1.a a + a + a = a 3a = a
Thus (S, +) is periodic
IV. CLASSES OF MULTIPLICATIVELY SUBIDEMPOTENT SEMIRING:
In a semiring S, an element a is Multiplicatively Subidempotent if a + a2
= a. A semiring S is
Multiplicatively Subidempotent if and only if each of its elements is Multiplicatively Subidempotent.
Multiplicatively Subidempotent semiring plays an important role in modal logic.
Lemma 4.1: Let S be a Multiplicatively Subidempotent semiring and „e‟ be additive and multiplicative identity.
Then S is an idempotent semiring.
Proof: By hypothesis e is an additive identity and also multiplicative identity then a.e = e.a = a and e + a = a
+ e = a for all a, e in S
Since S is multiplicatively subidempotent a + a2
= a for all a in S (1)
Equation (1) can be written as a (e + a) = a which implies a2
= a (2)
Thus (S, •) is a band
Adding a to both sides of equation (2) we obtain a + a2
= a + a
which implies a = a + a for all a in S (3)
From equations (2) and (3) we conclude that S is an idempotent semiring
Example 4.2: We have framed an example by considering the set S = {a, x} for above lemma which satisfies
all the conditions of lemma.
+ a x . a x
a a a a a a
x a x x a x
3. International Journal of Modern Research in Engineering and Technology (IJMRET)
www.ijmret.org Volume 1 Issue 2 ǁ July 2016.
w w w . i j m r e t . o r g Page 7
Theorem 4.3: Let S be a totally ordered multiplicatively subidempotent semiring. If S contains multiplicative
identity „1‟ in which additive identity ‟e‟ is also multiplicative identity and (S, +) and (S, •) are positively totally
ordered then S is a mono semiring and the addition and multiplication are given by a + b = b + a = ab = ba =
max (a, b).
Proof: By above lemma we have S is an idempotent semiring
Let a, b € S and a b implies a + a ≤ a + b ≤ b + b a ≤ a + b ≤ b
Since (S, +) is p.t.o this is possible only if a + b = b = max (a, b)
Also a b implies a2
≤ ab ≤ b2
which implies a ≤ ab ≤ b
Since (S, •) is p.t.o this is possible only if ab = b = max (a, b)
Example 4.4: Here (S, +) and (S, •) are p.t.o, y x a then a + b = b + a = ab = ba = max (a, b).
Example 4.5: Here (S, +) and (S, •) are n.t.o, a x y then a + b = b + a = ab = ba = min (a, b).
Note 4.6:
(i) In the above theorem if (S, +) and (S, •) are negatively totally ordered, then S is a mono semiring and
the addition and multiplication are given by a + b = b + a = ab = ba = min (a, b).
(ii) In a t.o.s.r, if (S, +) is p.t.o and (S, •) is n.t.o or vice-versa. We arrive contradiction to the hypothesis
that additive identity ‟e‟ is also multiplicative identity.
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+ a x y · a x y
a a a a a a a a
x a x x x a x x
y a x y y a x y
+ a x y · a x y
a a x y a a x y
x x x y x x x y
y y y y y y y y