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
Using ψ-Operator to Formulate a New Definition of Local Functionpaperpublications3
Abstract:In this paper, we use ψ-operator in order to get a new version of local function. The concepts of maps, dense, resolvable and Housdorff have been investigated in this paper, as well as modified to be useful in general الخلاصة: استخدمنافيهذاالبحثالعمليةابسايمنأجلالحصولعلىنسخهجديدهمنالدالةالمحلية.وقدتمالتحقيقفيمفاهيمالدوال, الكثافة,المجموعاتالقابلةللحلوفضاءالهاوسدورف,وكذلكتعديلهالتكونمفيدةبشكلعام. Keywords:ψ-operator, dense, resolvable and Housdorff.
Title:Using ψ-Operator to Formulate a New Definition of Local Function
Author:Dr.Luay A.A.AL-Swidi, Dr.Asaad M.A.AL-Hosainy, Reyam Q.M.AL-Feehan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
The document describes an algorithm for solving the Steiner Tree problem parameterized by treewidth. It uses dynamic programming over a tree decomposition of the input graph to compute the solution. The algorithm builds a dynamic programming table at each node of the tree decomposition. It handles leaf, forget, introduce and join nodes in t^O(t) time by updating/merging partial solutions. This results in an overall running time of t^O(t) to solve the Steiner Tree problem parameterized by treewidth.
Unit 1: Topological spaces (its definition and definition of open sets)nasserfuzt
Learning Objectives:
1. To understand the definition of topology with examples
2. To know the intersection and union of topologies
3. To understand the comparison of topologies
This document introduces some basic concepts of set theory, including:
1) Defining sets by listing elements or describing properties. Common sets include real numbers, integers, etc.
2) Basic set operations like union, intersection, difference, and complement.
3) Relationships between sets like subset, proper subset, and equality.
4) Other concepts like partitions, power sets, and Cartesian products involving ordered pairs from multiple sets.
The document discusses matroids and parameterized algorithms. It begins with an overview of Kruskal's greedy algorithm for finding a minimum spanning tree in a graph. It then introduces matroids and defines them as structures where greedy algorithms find optimal solutions. Several examples of matroids are provided, including uniform matroids, partition matroids, graphic matroids, and gammoids. The document also presents an alternate definition of matroids using basis families.
The document defines and discusses several key concepts relating to groups in abstract algebra:
- A group is defined as a non-empty set together with a binary operation that satisfies closure, associativity, identity, and inverse properties.
- An abelian (or commutative) group is one where the binary operation is commutative. Examples of abelian groups include integers under addition.
- The quaternion group is a non-abelian group of order 8 under multiplication.
- Theorems are presented regarding the uniqueness of identity and inverses in a group, as well as cancellation, reverse order, and inverse properties of groups.
Heuristics for counterexamples to the Agrawal ConjectureAmshuman Hegde
This document presents heuristics for constructing counterexamples to the Agrawal Conjecture. It generalizes an earlier proposition given by Lenstra and Pomerance by showing that their arguments can be applied to any prime number r that is congruent to 1 modulo 4. A second generalization is presented that allows the number n to be composed of prime power factors rather than just prime factors. Finally, a rough estimate is given suggesting that there should be at least e^{T^2(1-5/m)} counterexamples below e^{T^2} for large T.
Using ψ-Operator to Formulate a New Definition of Local Functionpaperpublications3
Abstract:In this paper, we use ψ-operator in order to get a new version of local function. The concepts of maps, dense, resolvable and Housdorff have been investigated in this paper, as well as modified to be useful in general الخلاصة: استخدمنافيهذاالبحثالعمليةابسايمنأجلالحصولعلىنسخهجديدهمنالدالةالمحلية.وقدتمالتحقيقفيمفاهيمالدوال, الكثافة,المجموعاتالقابلةللحلوفضاءالهاوسدورف,وكذلكتعديلهالتكونمفيدةبشكلعام. Keywords:ψ-operator, dense, resolvable and Housdorff.
Title:Using ψ-Operator to Formulate a New Definition of Local Function
Author:Dr.Luay A.A.AL-Swidi, Dr.Asaad M.A.AL-Hosainy, Reyam Q.M.AL-Feehan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
The document describes an algorithm for solving the Steiner Tree problem parameterized by treewidth. It uses dynamic programming over a tree decomposition of the input graph to compute the solution. The algorithm builds a dynamic programming table at each node of the tree decomposition. It handles leaf, forget, introduce and join nodes in t^O(t) time by updating/merging partial solutions. This results in an overall running time of t^O(t) to solve the Steiner Tree problem parameterized by treewidth.
Unit 1: Topological spaces (its definition and definition of open sets)nasserfuzt
Learning Objectives:
1. To understand the definition of topology with examples
2. To know the intersection and union of topologies
3. To understand the comparison of topologies
This document introduces some basic concepts of set theory, including:
1) Defining sets by listing elements or describing properties. Common sets include real numbers, integers, etc.
2) Basic set operations like union, intersection, difference, and complement.
3) Relationships between sets like subset, proper subset, and equality.
4) Other concepts like partitions, power sets, and Cartesian products involving ordered pairs from multiple sets.
The document discusses matroids and parameterized algorithms. It begins with an overview of Kruskal's greedy algorithm for finding a minimum spanning tree in a graph. It then introduces matroids and defines them as structures where greedy algorithms find optimal solutions. Several examples of matroids are provided, including uniform matroids, partition matroids, graphic matroids, and gammoids. The document also presents an alternate definition of matroids using basis families.
The document defines and discusses several key concepts relating to groups in abstract algebra:
- A group is defined as a non-empty set together with a binary operation that satisfies closure, associativity, identity, and inverse properties.
- An abelian (or commutative) group is one where the binary operation is commutative. Examples of abelian groups include integers under addition.
- The quaternion group is a non-abelian group of order 8 under multiplication.
- Theorems are presented regarding the uniqueness of identity and inverses in a group, as well as cancellation, reverse order, and inverse properties of groups.
Heuristics for counterexamples to the Agrawal ConjectureAmshuman Hegde
This document presents heuristics for constructing counterexamples to the Agrawal Conjecture. It generalizes an earlier proposition given by Lenstra and Pomerance by showing that their arguments can be applied to any prime number r that is congruent to 1 modulo 4. A second generalization is presented that allows the number n to be composed of prime power factors rather than just prime factors. Finally, a rough estimate is given suggesting that there should be at least e^{T^2(1-5/m)} counterexamples below e^{T^2} for large T.
1) The document discusses parallel summable range symmetric matrices over an incline. It defines what it means for two matrices to be parallel summable and defines their parallel sum.
2) It proves several properties of parallel summable matrices, including that the sum and parallel sum of parallel summable range symmetric matrices are also range symmetric.
3) It establishes conditions under which the sum or parallel sum of two matrices will be range symmetric, such as if their row and column ranges are properly contained within each other.
This document summarizes key concepts in commutative ring theory, beginning with divisibility and ideals. It defines what it means for an element to divide another in a ring, as well as irreducible, prime, and unit elements. Ideals are introduced as additive subgroups that are closed under multiplication. Polynomial rings over a field are then discussed, proving they have unique factorizations. Irreducible polynomials are shown to be prime. The document provides numerous examples and proofs of fundamental results about divisibility, ideals, and factorization in commutative rings.
The document defines sets, elements, and operations on sets. It discusses that a set is a collection of objects, called elements or points. It provides examples of sets and defines subset and subset notation. The document then defines functions, including the domain, codomain, range, and image of a function. It discusses inverse functions and inverse image notation. Finally, it discusses characteristics functions, where the characteristics function of a set A maps elements in A to 1 and elements not in A to 0, completely describing the set.
In this paper, the concepts of sequences and series of complement normalized fuzzy numbers are introduced in terms of 𝛾-level, so that some properties and characterizations are presented, and some convergence theorems are proved
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
This document discusses ideals and factor rings. It begins by defining an ideal as a subring of a ring where certain multiplication rules hold. It provides examples of ideals in various rings. It then discusses the existence of factor rings when a subring is an ideal. The elements of a factor ring are cosets of the ideal. It provides examples of computing operations in specific factor rings. Finally, it discusses prime and maximal ideals and their relationship to the structure of factor rings.
The document discusses the key topics in discrete mathematics that will be covered across 5 units. Unit 1 covers sets, relations, functions and their properties. Unit 2 discusses mathematical induction, counting techniques, and number theory topics. Unit 3 is on propositional logic, logical equivalence and proof techniques. Unit 4 covers algebraic structures like groups, rings and fields. Unit 5 is about graphs, trees, and their properties like coloring and shortest paths. The document also lists 3 recommended textbooks for the course.
Abstract Quadripartitioned single valued neutrosophic (QSVN) set is a powerful structure where we have four components Truth-T, Falsity-F, Unknown-U and Contradiction-C. And also it generalizes the concept of fuzzy, initutionstic and single valued neutrosophic set. In this paper we have proposed the concept of K-algebras on QSVN, level subset of QSVN and studied some of the results. In addition to this we have also investigated the characteristics of QSVN Ksubalgebras under homomorphism.
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
1) Predicates become propositions when variables are quantified by assigning values or using quantifiers. Quantifiers like ∀ and ∃ are used to make statements true or false for all or some values.
2) ∀ (universal quantifier) means "for all" and makes a statement true for all values of a variable. ∃ (existential quantifier) means "there exists" and makes a statement true if it is true for at least one value.
3) Predicates with unbound variables are neither true nor false. Binding variables by assigning values or using quantifiers turns predicates into propositions that can be evaluated as true or false.
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
An Overview of Separation Axioms by Nearly Open Sets in Topology.IJERA Editor
Abstract: The aim of this paper is to exhibit the research on separation axioms in terms of nearly open sets viz
p-open, s-open, α-open & β-open sets. It contains the topological property carried by respective ℘ -Tk spaces (℘
= p, s, α & β; k = 0,1,2) under the suitable nearly open mappings . This paper also projects ℘ -R0 & ℘ -R1
spaces where ℘ = p, s, α & β and related properties at a glance. In general, the ℘ -symmetry of a topological
space for ℘ = p, s, α & β has been included with interesting examples & results.
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignment.
This document discusses potentialist modal set theory and reflection principles. It proposes a modal logic called L∈ that adds modal operators to the language of set theory. Potentialist modal set theory (MST) is defined using priority and plenitude axioms formulated in L∈. MST corresponds to Zermelo-Fraenkel set theory plus rank-into-rank when augmented with complete reflection. Weaker reflection principles like partial reflection correspond to weaker theories. Potentialist reflection principles can be formulated without reference to sets and may apply more broadly than set theory.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
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.
The EM algorithm is used to find maximum likelihood estimates for problems with latent variables. It works by alternating between an E-step (computing expected values of the latent variables) and an M-step (maximizing the likelihood with respect to the parameters). For mixture of Gaussians, the E-step computes the posterior probabilities that each data point belongs to each component. The M-step then updates the mixture weights, means, and covariances by taking weighted averages/sums of the data using these posteriors.
This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can express properties and relationships for objects. Laws of quantifier equivalence are also presented.
This document discusses predicates and quantifiers in predicate logic. It begins by explaining the limitations of propositional logic in expressing statements involving variables and relationships between objects. It then introduces predicates as statements involving variables, and quantifiers like universal ("for all") and existential ("there exists") to express the extent to which a predicate is true. Examples are provided to demonstrate how predicates and quantifiers can be used to represent statements and enable logical reasoning. The document also covers translating statements between natural language and predicate logic, and negating quantified statements.
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.
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.
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
On fixed point theorems in fuzzy 2 metric spaces and fuzzy 3-metric spacesAlexander Decker
1) The document discusses fixed point theorems for mappings in fuzzy 2-metric and fuzzy 3-metric spaces.
2) It defines concepts like fuzzy metric spaces, Cauchy sequences, compatible mappings, and proves some fixed point theorems for compatible mappings.
3) The theorems show that under certain contractive conditions on the mappings, there exists a unique common fixed point for the mappings in a complete fuzzy 2-metric or fuzzy 3-metric space.
1) The document discusses parallel summable range symmetric matrices over an incline. It defines what it means for two matrices to be parallel summable and defines their parallel sum.
2) It proves several properties of parallel summable matrices, including that the sum and parallel sum of parallel summable range symmetric matrices are also range symmetric.
3) It establishes conditions under which the sum or parallel sum of two matrices will be range symmetric, such as if their row and column ranges are properly contained within each other.
This document summarizes key concepts in commutative ring theory, beginning with divisibility and ideals. It defines what it means for an element to divide another in a ring, as well as irreducible, prime, and unit elements. Ideals are introduced as additive subgroups that are closed under multiplication. Polynomial rings over a field are then discussed, proving they have unique factorizations. Irreducible polynomials are shown to be prime. The document provides numerous examples and proofs of fundamental results about divisibility, ideals, and factorization in commutative rings.
The document defines sets, elements, and operations on sets. It discusses that a set is a collection of objects, called elements or points. It provides examples of sets and defines subset and subset notation. The document then defines functions, including the domain, codomain, range, and image of a function. It discusses inverse functions and inverse image notation. Finally, it discusses characteristics functions, where the characteristics function of a set A maps elements in A to 1 and elements not in A to 0, completely describing the set.
In this paper, the concepts of sequences and series of complement normalized fuzzy numbers are introduced in terms of 𝛾-level, so that some properties and characterizations are presented, and some convergence theorems are proved
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
This document discusses ideals and factor rings. It begins by defining an ideal as a subring of a ring where certain multiplication rules hold. It provides examples of ideals in various rings. It then discusses the existence of factor rings when a subring is an ideal. The elements of a factor ring are cosets of the ideal. It provides examples of computing operations in specific factor rings. Finally, it discusses prime and maximal ideals and their relationship to the structure of factor rings.
The document discusses the key topics in discrete mathematics that will be covered across 5 units. Unit 1 covers sets, relations, functions and their properties. Unit 2 discusses mathematical induction, counting techniques, and number theory topics. Unit 3 is on propositional logic, logical equivalence and proof techniques. Unit 4 covers algebraic structures like groups, rings and fields. Unit 5 is about graphs, trees, and their properties like coloring and shortest paths. The document also lists 3 recommended textbooks for the course.
Abstract Quadripartitioned single valued neutrosophic (QSVN) set is a powerful structure where we have four components Truth-T, Falsity-F, Unknown-U and Contradiction-C. And also it generalizes the concept of fuzzy, initutionstic and single valued neutrosophic set. In this paper we have proposed the concept of K-algebras on QSVN, level subset of QSVN and studied some of the results. In addition to this we have also investigated the characteristics of QSVN Ksubalgebras under homomorphism.
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
1) Predicates become propositions when variables are quantified by assigning values or using quantifiers. Quantifiers like ∀ and ∃ are used to make statements true or false for all or some values.
2) ∀ (universal quantifier) means "for all" and makes a statement true for all values of a variable. ∃ (existential quantifier) means "there exists" and makes a statement true if it is true for at least one value.
3) Predicates with unbound variables are neither true nor false. Binding variables by assigning values or using quantifiers turns predicates into propositions that can be evaluated as true or false.
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
An Overview of Separation Axioms by Nearly Open Sets in Topology.IJERA Editor
Abstract: The aim of this paper is to exhibit the research on separation axioms in terms of nearly open sets viz
p-open, s-open, α-open & β-open sets. It contains the topological property carried by respective ℘ -Tk spaces (℘
= p, s, α & β; k = 0,1,2) under the suitable nearly open mappings . This paper also projects ℘ -R0 & ℘ -R1
spaces where ℘ = p, s, α & β and related properties at a glance. In general, the ℘ -symmetry of a topological
space for ℘ = p, s, α & β has been included with interesting examples & results.
Fixed Point Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignment.
This document discusses potentialist modal set theory and reflection principles. It proposes a modal logic called L∈ that adds modal operators to the language of set theory. Potentialist modal set theory (MST) is defined using priority and plenitude axioms formulated in L∈. MST corresponds to Zermelo-Fraenkel set theory plus rank-into-rank when augmented with complete reflection. Weaker reflection principles like partial reflection correspond to weaker theories. Potentialist reflection principles can be formulated without reference to sets and may apply more broadly than set theory.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
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.
The EM algorithm is used to find maximum likelihood estimates for problems with latent variables. It works by alternating between an E-step (computing expected values of the latent variables) and an M-step (maximizing the likelihood with respect to the parameters). For mixture of Gaussians, the E-step computes the posterior probabilities that each data point belongs to each component. The M-step then updates the mixture weights, means, and covariances by taking weighted averages/sums of the data using these posteriors.
This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can express properties and relationships for objects. Laws of quantifier equivalence are also presented.
This document discusses predicates and quantifiers in predicate logic. It begins by explaining the limitations of propositional logic in expressing statements involving variables and relationships between objects. It then introduces predicates as statements involving variables, and quantifiers like universal ("for all") and existential ("there exists") to express the extent to which a predicate is true. Examples are provided to demonstrate how predicates and quantifiers can be used to represent statements and enable logical reasoning. The document also covers translating statements between natural language and predicate logic, and negating quantified statements.
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.
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.
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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.
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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
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Mathematics subject classifications: 45H10, 54H25
The maiin objjectt off tthiis paper tto iinttroduce T--pre--operattor,,T--pre--open sett,, T--pre--monottone,,pre--
subadiittiive operattor and pre--regullar operattor.. As wellll as we iinttroduce ((T,,L))pre--conttiinuiitty.
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P-Pseudo Symmetric Ideals in Ternary Semiring
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. V (Jan - Feb. 2015), PP 105-111
www.iosrjournals.org
DOI: 10.9790/5728-1115105111 www.iosrjournals.org 105 | Page
P-Pseudo Symmetric Ideals in Ternary Semiring
Dr. D. MadhusudhanaRao1
, G. SrinivasaRao2
1
Lecturer and Head, Department of Mathematics, VSR & NVR College, Tenali, A. P. India.
2
Asst. Professor, Department of Mathematics, Tirumala Engineering College, Narasarao Pet, A. P. India.
Abstract: 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 P-
pseudo 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.
Mathematical Subject Classification: 16Y30, 16Y99.
Key Words: Pseudo Symmetric ideal, P-pseudo symmetric ideal, P-Prime, Completely P-Prime, P-Semiprime,
Completely P-Semiprime, Pseudo Commutative, mid unit.
I. Introduction:
Algebraic structures play a prominent role in mathematics with wide ranging applications in many
disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding
theory, topological spaces, and the like. The theory of ternary algebraic systems was introduced by D. H.
Lehmer [5]. He investigated certain ternary algebraic systems called triplexes which turn out to be commutative
ternary groups. After that W. G. Lister[6] studied about ternary semirings. About T. K. Dutta and S. Kar [1, 3]
introduced and studied some properties of ternary semirings which is a generalization of ternary rings. Dheena.
P, Manvisan. S [2] made a study on P-prime and small P-prime ideals in semirings. S. Kar [4] investigated on
quasi ideals and bi-ideals in ternary semirings.
D. Madhusudhana Rao, A. Anjaneyulu and A. Gangadhara Rao [7] in 2011 introduced the notion of
pseudo symmetric ideals in Γ-Semigroups. In 2012 Y. Sarala, A. Anjaneyulu and D. Madhusudhana Rao [13]
introduced the same concept to the ternary semigroups. In 2014 D. MadhusudhanaRao and G. Srinivasa Rao [8,
9] investigated and studied about classification of ternary semirings and some special elements in a ternary
semirings. D. Madhsusudhana Rao and G. Srinivasa Rao [10] introduced and investigated structure of certain
ideals in ternary semirings. D. Madhusudhana Rao and G. Srinivasa Rao[11] also introduced the structure of
completely P-prime, P-prime, Completely P-Semiprime and P-semiprime ideals in Ternary semirings. After
that they [12] made a study and investigated prime radicals in ternary semitings. Our main purpose in this paper
is to introduce the Structure of P-pseudo symmetric Ideals in ternary semirings.
II. Preliminaries:
Definition II.1[6] : A nonempty set T together with a binary operation called addition and a ternary
multiplication denoted by [ ] is said to be a ternary semiring if T is an additive commutative semigroup
satisfying the following conditions :
i) [[abc]de] = [a[bcd]e] = [ab[cde]],
ii) [(a + b)cd] = [acd] + [bcd],
iii) [a(b + c)d] = [abd] + [acd],
iv) [ab(c + d)] = [abc] + [abd] for all a; b; c; d; e ∈T.
Throughout Twill denote a ternary semiring unless otherwise stated.
Note II.2 : For the convenience we write 1 2 3x x x instead of 1 2 3x x x
Note II.3 : Let T be a ternary semiring. If A,B and C are three subsets of T , we shall denote the set
ABC = : , ,abc a A b B c C .
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Note II.4 : Let T be a ternary semiring. If A,B are two subsets of T , we shall denote the set
A + B = : ,a b a A b B .
Note II.5 : Any semiring can be reduced to a ternary semiring.
Example II.6 [6] :Let T be an semigroup of all m × n matrices over the set of all non negative rational numbers.
Then T is a ternary semiring with matrix multiplication as the ternary operation.
Example II.7 [6]:Let S = {...,−2i, −i, 0, i, 2i, ...} be a ternary semiring withrespect to addition and complex
triple multiplication.
Definition II.8 [7]:An element a of a ternary semiring T is said to be a mid-unit provided xayaz = xyz for allx,
y, z∈ T.
Definition II.9 [6]: A ternary semiring T is said to be commutative ternary semiring providedabc = bca = cab
= bac = cba = acbfor all a,b,c T.
Definition II.10 [6]: A ternary semiring T is said to be left pseudo commutative provided
abcde = bcade = cabde = bacde = cbade = acbde a,b,c,d,eT.
Definition II.11 [6] : A ternary semiring T is said to be a lateral pseudo commutativeternary semiring provide
abcde = acdbe = adbce = acbde = adcbe = abdcefor alla,b,c,d,eT.
Definition II.12 [6]: A ternary semiring T is said to be right pseudo commutative provided abcde = abdec =
abecd = abdce = abedc = abced a,b,c,d,eT.
Definition II.13 [6]: A ternary semiring T is said to be pseudo commutative, provided T is a left pseudo
commutative, right pseudo commutative and lateral pseudo commutative ternary semiring.
Definition II.14 [8]: A nonempty subset A of a ternary semiring T is said to be ternary ideal or simply an
ideal of T if
(1) a, b ∈ A implies a + b∈ A
(2) b, c T , a A implies bca A, bacA, abcA.
Definition II.15 [9]:An ideal A of a ternary semiring T is said to be a completely prime ideal of T provided
x, y, z T and xyzA implies either x A or yA or zA.
Definition II.16 [9]:An ideal A of a ternary semiring T is said to be a completely P-prime ideal of T provided
x, y, z T and xyz+ P⊆A implies either x A or yA or zA for any ideal P.
Definition II.17 [9]: An ideal A of a ternary semiring T is said to be a P-prime ideal of T provided X,Y,Z are
ideals of T and XYZ + P A X A or Y A or Z A for any ideal P.
Theorem II.18[10]: Every completely P-prime ideal of a ternary semiring T is a P-prime ideal of T.
Theorem II.19[10] : Every completely P-semiprime ideal of a ternary semiring T is a P-semiprime ideal of
T.
Definition II.20 [9]: An ideal A of a ternary semiring T is said to be a completely P-semiprime ideal provided
xT, n
x p A for some odd natural number n>1 and p ∈ P implies xA.
Definition II.21 [9]: An ideal A of a ternary semiring T is said to be semiprime ideal provided X is an ideal of
T and Xn
A for some odd natural number nimplies X ⊆ A.
Definition 2.22 [9]: An ideal A of a ternary semiring T is said to be P-semiprime ideal provided X is an ideal
of T and Xn
+ P A for some odd natural number nimplies X ⊆ A.
Notation II.23 [11]: If A is an ideal of a ternary semiring 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 II.24 [11]: If A is an ideal of a ternary semiring T, then A 4A 3A 2A 1A .
III. P-Pseudo Symmetric Ideals In Ternary Semirings
We now introduce the notion of a pseudo symmetric ideal of a ternary semiring.
Definition III.1 : An ideal A of a ternary semiring T is said to be pseudo symmetric provided x, y, z T,
xyzA implies xsytzA for all s, tT.
We now introduce the notion of a P-pseudo symmetric ideal of a ternary semiring.
Definition III.2 : A pseudo symmetric ideal A of a ternary semiring T is said to be
P-pseudo symmetric ideal provided x, y, z T and P is an ideal of T, xyz + pA implies xsytz + p A for
all s, tT and p ∈ P.
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Note III.3 : A pseudo symmetric ideal A of a ternary semiring T is said to be P-pseudo symmetric ideal
provided x, y, z T and P is an ideal of T, xyz + P ⊆ A implies xsytz + P ⊆A for all s, tT.
Theorem III.4 : Let A be a pseudo symmetric ideal in a ternary semiring T and ai, bi, ciT. Then
1
n
i i i
i
a bc
A if and only if < a > < b > < c > A.
Proof : Suppose < a > < b > < c > A.
1
n
i i i
i
a bc
< a > < b > < c >A
1
n
i i i
i
a bc
A.
Conversely suppose that
1
n
i i i
i
a bc
A. Let t < a > < b > < c>.
Then t = s1a1s2b1s3c1s4where s1 ,s2 , s3, s4T1
.
a1b1c1A, s2, s3 T1
, A is pseudo symmetric ideal
a1s2b1s3c1As1a1s2b1s3c1s4A t A. Therefore < a > < b > < c > A.
Corollary III.5 : Let A be any pseudo symmetric ideal in a ternary semiring T and a1, a2….,an T where
n is an odd natural number. Then a1a2….an A if and only if < a1>< a2>….<an> A.
Proof : Clearly if < a1>>< a2>….<an>A, thena1a2….anA where n is an odd natural number.
Conversely suppose that a1a2….anA where n is an odd natural number.
Let t<a1>< a2>….<an>. Then t =s1a1 s2a2….ansn+1, where siTe
, i =1,2,…n+1.
a1a2….anA, A is pseudo symmetric ideal s1a1 s2a2….ansn+1A and hence tA.
Therefore <a1>< a2>….<an> A.
Theorem III.6 : Let A be a P-pseudo symmetric ideal in a ternary semiring T and ai, bi, ciT and p ∈ P.
Then
1
n
i i i
i
a bc p
∈ A if and only if < a > < b > < c > + P A.
Proof : Suppose <a > < b > < c > + PA.
Then
1
n
i i i
i
a bc p
< a > < b > < c > + PA
1
n
i i i
i
a bc p
A.
Conversely suppose that
1
n
i i i
i
a bc p A
. Let t < a > < b > < c > + P.
Then t = s1a1s2b1s3c1s4 + p where s1 ,s2 , s3, s4T1
, p ∈ P.
a1b1c1 + p A, s2, s3 T1
, A is P-pseudo symmetric ideal
a1s2b1s3c1 + p A. Then a1s2b1s3c1A and p ∈ A.
s1a1s2b1s3c1s4A and p ∈ A s1a1s2b1s3c1s4 + p ∈ A
⇒ t A. Therefore < a > < b > < c > + P A.
Corollary III.7 : Let A be any P-pseudo symmetric ideal in a ternary semiring T and a1, a2….,an T
where n is an odd natural number and p ∈ P. Then a1a2….an + p A if and only if < a1> < a2>….<an> +
P A.
Proof : Clearly if < a1> < a2>….< an > + P A, then < a1> < a2>….< an > A, P A ⇒ a1a2….anA where
n is an odd natural number and p ∈ A for all p ∈ A ⇒ a1a2….an + p ∈ A .
Conversely suppose that a1a2….an + p A where n is an odd natural number and p ∈ P.
Let t< a1> < a2>….< an> + P. Then t =s1a1 s2a2….ansn+1 + p, where siTe
, i =1,2,…n+1,
p ∈ P. a1a2….anA, p ∈ P and A is P-pseudo symmetric ideal s1a1 s2a2….ansn+1 + pA and hence tA.
Therefore <a1>< a2>….<an> + P A.
Corollary III.8: Let A be a P-pseudo symmetric ideal in a ternary semiring T, then for any odd natural
number n, an
+ pA implies < a >n
+ PA.
Proof : The proof follows from corollary 4.1.7, by taking a1=a2=….an=a.
Corollary III.9 : Let A be a P-pseudo symmetric ideal in a ternary semiring T. If an
A, for some odd
natural number n then < ast >n
+ P, < sta >n
+ P, < sat >n
+ P ⊆ A for all s, t∈ T and for some ideal P.
Theorem 4.1.10 : Every completely P-semiprime ideal A in a ternary semiring T is a P-pseudo symmetric
ideal.
Proof : Let A be a completely P-semiprime ideal of the ternary semiring T.
Let x, y, zT, p ∈ P and xyz + p A ⇒ xyzA, p ∈ A. xyzA implies (yzx)3
= (yzx)(yzx)(yzx) =
yz(xyz)(xyz)xA and p ∈ A.
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(yzx)3
+ p A, A is completely P-semiprime ideal yzxA.
Similarly (zxy)3
+ p = (zxy)(zxy)(zxy) + p = z(xyz)(xyz)xy + p ∈ A.
(zxy)3
+ p ∈ A, A is completely P-semiprime ideal ⇒zxy∈ A.
If s, t T1
, then (xsytz)3
+ p= (xsytz)(xsytz) (xsytz) + p = xsyt[zx(syt)(zxs)y]tz + p A.
(xsytz)3
+ p A, A is completely P-semiprime ⇒ xsytzA.
Therefore A is a P-pseudo symmetric ideal.
Note III.11 : The converse of theorem 4.1.9, is not true, i.e., a P-pseudo symmetric ideal of a ternary semiring
need not be completely P-semiprime.
Example III.12 : Let T = {a, b, c} and P = {a}. Define a ternary operation [ ] on T as [abc] = a.b.c where . is
binary operation and the binary operation defined as follows
Clearly (T, +, [ ] ) is a ternary semiring and {a}, {a, b}, T are the ideals of T.
Now aaa + a{a} aaaaa + a, ababa + a, acaca + a, aaaba + a, abaca + a,
acaba + a{a}
abb + a{a}aabab + a, abbbb + a, acbcb + a, aabbb + a, abbcb + a, acbab + a {a}
baa + a{a}baaaa + a, bbbba + a, bcaca + a, baaba + a, bbaca + a, bcaaa + a {a}
aba + a ∈{a}⇒aabaa + a, abbba + a, acbca + a, aabba + a, abbca + a, acbaa + a ∈ {a}
bab + a ∈{a}⇒baaab + a, bbabb + a, bcbcb + a, baabb + a, bbacb + a, bcaab + a ∈ {a}
bbb + a ∈{a}⇒babab + a, bbbbb + a, bcbcb + a, babbb + a, bbbcb + a, bcbab + a ∈ {a}
bba + a ∈{a}⇒babaa + a, bbbba + a, bcbca + a, babba + a, bbbca + a, bcbaa + a ∈ {a}
acc + a ∈{a}⇒ aacac + a, abcbc + a, acccc + a, aacbc + a, abccc + a, accac + a ∈ {a}
caa + a {a}caaaa + a, cbcba + a, ccaca + a, caaba + a, cbaca + a, ccaaa + a {a}
aca + a ∈{a}⇒ aacaa + a, abcba + a, accca + a, aacba + a, abcca + a, accaa + a ∈ {a}
cac + a ∈{a}⇒caaac + a, cbabc + a, ccacc + a, caabc + a, cbacc + a, ccaac + a ∈ {a}
cca + a ∈{a}⇒cacaa + a, cbcba + a, cccca + a, cacba + a, cbcca + a, cccaa + a ∈ {a}
abc + a {a} aabac + a, abbbc + a, acbcc + a, aabbc + a, abbcc + a, acbac + a {a}
bca + a {a} bacaa + a, bbcba + a, bccca + a, bacba + a, bbcca + a, bccaa + a {a}
cab + a {a} caaab + a, cbabb + a, ccacb + a, caabb + a, abacb + a, ccaab + a {a}.
Therefore {a} is a P-pseudo symmetric ideal in T. Here b3
+ a = a{a}, but b{a}.
Therefore {a} is not a completely P-semiprime ideal.
Theorem III.13 : If A is a P-pseudo symmetric ideal of a ternary semiring T then A2 = A4.
Proof : By theorem II.24, A4 A2. Let xA2 .
Then xn
A for some odd natural number n.
Since A is P-pseudo symmetric, xn
+ pA < x >n
+ PA< x >n
⊆ A and P ⊆ A
⇒ < x >n
⊆ A ⇒ x ∈ A4. Therefore A2 A4 and hence A2 = A4.
Theorem III.14: If A is a P-pseudo symmetric ideal of a ternary semiring T then A2 = {x : xn
∈ A for some
odd natural number n∈ N} is a minimal completely P-semiprme ideal of T.
Proof : Clearly A A2 and hence A2 is a nonempty subset of T. Let x A2 and s, tT.
Now x A2 xn
A for some odd natural number n. xn
A, s, tT, A is a P-pseudo symmetric ideal of T
(xst)n
+ pA, (sxt)n
+ p ∈ A, (stx)n
+ p A for p ∈ P
(xst)n
, (sxt)n
, (stx)n
A, pA ⇒ xst, sxt, stx A2.
Therefore A2 is an ideal of T. Let xT and x3
+ p A2.
Now x3
+ p A2 x3
∈ A2 and p ∈ A2 ⇒ x3
∈ A2 ⇒(x3
)n
A for some odd natural number n
x3n
Ax A2. So A2 is a completely P-semiprime ideal of T.
Let Q be any completely P-semiprime ideal of T containing A. Let x A2.
Then xn
A for some odd natural number n. By corollary 4.1.7, xn
+ pA
< x >n
+ P A Q. Since Q is completely P-semiprime, < x >n
+ P QxQ.
Therefore A2 is the minimal completely P-semiprime ideal of T containing A.
Theorem III.15 : If A is a P-pseudo symmetric ideal of a ternary semiring T then
A4 = {x : < x >n
A for some odd natural number n} is the minimal P-semiprime ideal of T containing A.
+ a b c
a a b c
b b b c
c c c c
. a b c
a a a a
b a a a
c a b c
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Proof : Clearly A A4 and hence A4 is a nonempty subset of T. Let x A4 and s, tT.
Since x A4, < x >n
A for some odd natural number n.
Now < xst >n
<x>n
A, < sxt >n
and < stx >n
< x >n
A xst, sxt, stx A4.
Then A4 is an ideal of T containing A.
Let xT such that < x >3
+ P A4 ⇒ < x >3
⊆ A4, P ⊆ A4.
Then < x >3
⊆ A4 ⇒ ( <x>3
)n
A <x>3n
A x A4. Therefore A4 is a P-semiprime ideal of T
containing A. Suppose Q is a P-semiprime ideal of T containing A.
Let x A4. Then < x >n
A Q for some odd natural number n.
Since Q is a P-semiprime ideal of T, < x >n
+ PQ for some odd natural number nx Q. Therefore
A4 Q and hence A4 is the minimal P-semiprime ideal of T containing A.
Theorem III.16 : Every P-prime ideal Q minimal relative to containing a P-pseudo symmetric ideal A in a
ternary semiring T is completely P-prime.
Proof : Let S be a ternary sub semiring generated by TQ. First we show that A∩S = ∅. If A∩S ≠ ∅, then there
exist x1, x2, x3,……., xn∈ TQ such that x1x2x3…….xn + p ∈ A where n is an odd natural number and p ∈ P. By
corollary 4.1.7, < x1>< x2> ………… < xn> + P ⊆ A ⊆ Q. Since Q is a P-prime ideal, we have < xi>⊆ Q for
some i. It is a contradiction. Thus A∩S=∅. Consider the set Σ = { B : B is an ideal in T containing A such that
B∩S = ∅}. Since A ∈ Σ, Σ is nonempty. Now Σ is a poset under set inclusion and satisfies the hypothesis of
Zorn’s lemma. Thus by Zorn’s lemma, Σ contains a maximal element, say M. Let X, Y and Z be three ideals in
T such that XYZ + P ⊆ M. If X ⊈M , Y ⊈ M, Z ⊈ M, then M ∪X, M ∪ Y and M ∪ Z are ideals in T containing
M properly and hence by the maximality of M, we have (M∪X)∩S≠∅, (M ∪Y) ∩ S ≠ ∅ and(M ∪Z) ∩ S ≠ ∅.
Since M ∩ S = ∅, we have X ∩ S ≠ ∅, Y ∩ S ≠ ∅ and Z ∩ S ≠ ∅. So there exists x ∈ X ∩ S , y ∈ Y ∩ S and z ∈
Z ∩ S . Now, xyz ∈ XYZ ∩ T ⊆ M ∩T = ∅. It is a contradiction. Therefore either X ⊆M or Y ⊆ M or Z ⊆ M
and hence M is P-prime ideal containing A. Now, A ⊆M ⊆TS ⊆P. Since Q is a minimal P-prime ideal relative
to containing A, we have M = TS = Q and S = TQ. Let xyz + p ∈ Q. Then xyz ∉ S. Suppose if possible x ∉ Q,
y ∉ Q, z∉ Q. Now x ∉ Q, y ∉ Q, z ∉ Q ⇒ x, y, z ∈TQ ⇒x, y, z ∈S ⇒ xyz ∈ S. It is a contradiction. Therefore
either x ∈ Q or y ∈ Q or z ∈ Q. Therefore Q is a completely P-prime ideal of T.
Theorem III.17 : Let A be an ideal of a ternary semiring T. Then A is completely P-prime iff A is P-
prime and P-pseudo symmetric.
Proof : Suppose A is a completely P-prime ideal of T. By theorem II.18, A is P-prime.
Let x, y, z ∈ T, p ∈ P and xyz + p ∈ A.
xyz + p ∈ A, A is completely P-prime ⇒ x ∈ A or y ∈ A or z ∈ A
⇒xsytz + p ∈ A for all s, t ∈ T p ∈ P. Therefore A is a P-pseudo symmetric ideal.
Conversely Suppose that A is P-prime and P-pseudo symmetric.
Let x, y, z ∈ T, p ∈ P and xyz + p ∈ A. xyz + p ∈ A, A is a P-pseudo symmetric ideal
⇒< x > < y > < z > + P ⊆ A ⇒ < x > < y > < z > ⊆ A and P ⊆ A
⇒ < x > ⊆ A or < y > ⊆ A or < z >⊆A ⇒x ∈ A or y ∈ A or z ∈ A.
Therefore A is completely P-prime.
Corollary III.18: Let A be an ideal of a ternary semiring T. Then A is completely P-prime iff A is P-
prime and completely P-semiprime.
Proof : The proof follows from theorem III.17,
Corollary III.19 : Let A be an ideal of a ternary semiring T. Then A is completely P-semiprime iff A is P-
semiprime and P-pseudo symmetric.
Proof : Suppose that A is completely P-semiprime. By theorem II.19, A is P-semiprime and also by theorem
III.10, A is P-pseudo symmetric.
Conversely suppose that A is P-semiprime and P-pseudo symmetric.
Let x ∈ T, p ∈ P and x3
+ p∈ A. x3
+ p ∈ A, A is P-pseudo symmetric
⇒ <x3
> + P ⊆ A ⇒ < x > ⊆ A ⇒ x∈ A.
Therefore A is a completely P-semiprime ideal of T.
Theorem III.20 : Let A be a P-pseudo symmetric ideal of a ternary semiring T. Then the following are
equivalent.
1) A1 = The intersection of all completely prime ideals of T containing A.
2)
1
1A = The intersection of all minimal completely prime ideals of T containing A.
3)
11
1A = The minimal completely semiprime ideal of T relative to containing A.
4) A2 = { x ∈ T : xn
∈ A for some odd natural number n}
5) A3 = The intersection of all prime ideals of T containing A.
6)
1
3A = The intersection of all minimal prime ideals of T containing A.
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7) 11
3A = The minimal semiprime ideal of T relative to containing A.
8) A4 = {x ∈ T : < x >n
⊆ A for some odd natural number n}
Proof: Since completely P-prime ideals containing A and minimal completely prime ideals containing A and
minimal completely semiprime ideal relative to containing A are coincide, it follows that A1= 1
1A = 11
1A . Since
prime ideals containing A and minimal prime ideals containing A and the minimal semiprime ideal relative to
containing A are coincide, it follows that 3A = 1
3A = 11
3A . Since A is pseudo symmetric ideal, we have A2=A4.
Now by corollary III.15, we have 11
1A = 11
3A . Therefore A1= 1
1A = 11
1A = 3A = 1
3A = 11
3A and A2=A4. Hence the
given conditions are equivalent.
Definition III.21 : A ternary semiring T is said to be a pseudo symmetric ternary semiring provided every
ideal in T is a pseudo symmetric ideal.
Definition III.22 : A ternary semiring T is said to be a P-pseudo symmetric ternary semiring provided every
ideal in T is a P-pseudo symmetric ideal.
Theorem III.23 : Every commutative ternary semiring is a pseudo symmetric ternary semiring.
Proof : Suppose T is commutative ternary semiring.
Then abc = bca = cab = bac = cba = acb for all a, b, c ∈ T. Let A be an ideal of T.
Let a, b, c ∈ T, abc ∈ A and s, t ∈ T. Then asbtc = abstc = absct = abcst ∈ A.
Therefore A is a pseudo symmetric ideal and hence T is a pseudo symmetric ternary semiring.
Theorem III.24 : Every commutative ternary semiring is a P-pseudo symmetric ternary semiring.
Proof : Suppose T is commutative ternary semiring.
Then abc = bca = cab = bac = cba = acb for all a, b, c∈ T. Let A be an ideal of T.
Let a, b, c ∈ T and p ∈ P where P is any ideal of A, abc + p ∈ A.
⇒ abc ∈ A and p ∈ A. abc ∈ A and s, t ∈ T. Then asbtc = abstc = absct = abcst ∈ A
⇒ asbtc + p = abstc + p = absct + p = abcst + p ∈ A
Therefore A is a P-pseudo symmetric ideal and hence T is a P-pseudo symmetric ternary semiring.
Theorem III.25 : Every pseudo commutative ternary semiring is a P-pseudo symmetric ternary semiring.
Proof : Let T be a pseudo commutative ternary semiring and A be any ideal of T.
Let x, y, z ∈ T, xyz + P ⊆ A where P is any ideal. Then xyz ∈ A and P ⊆ A.
If s, t ∈ T. Then xsytz = syxtz = syzxt = s(xyz)t∈ A.
Therefore xsytz + P ⊆ A for all s, t ∈ T. Therefore A is a P-pseudo symmetric ideal.
Therefore T is a P-pseudo symmetric semiring.
Theorem III.26 : If T is a ternary semiring in which every element is a midunit then T is a P-pseudo
symmetric ternary semiring.
Proof : Let T be a ternary semiring in which every element is a midunit and A be any ideal of T. Let x, y, z ∈ T
and xyz + P ⊆ A where P is any ideal A. Then xyz ∈ A and P ⊆ A.
If s ∈ T, then s is a midunit and hence, xsysz = xyz ∈ A ⇒ xsysz + P ⊆ A. Hence A is a P-pseudo symmetric
ideal. Therefore T is a P-pseudo symmetric ternary semiring.
IV. Conclusion
In this paper mainly we studied about the P-pseudo symmetric ideals in ternary semirings.
Acknowledgment:
The authors would like to thank the referee(s) for careful reading of the manuscript.
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[4]. Kar, S., On quasi-ideals and bi- ideals in ternary semirings, Int. J. Math.Math.Sc., 18 (2005), 3015{3023.
[5]. Lehmer. D. H., A ternary analogue of abelian groups, Amer. J. Math., 59(1932), 329-338.
[6]. Lister, W.G., Ternary rings, Trans Amer. Math.Soc., 15IV (1971), 37- 55.
[7]. Madhusudhana Rao. D,Anjaneyoulu. A and Gangadhara Rao. A., Pseudo Symmetric Γ-ideals in Γ- Semigroups-Internationla
eJournals of Mathematics and Engineering, 116(2011), 1074-1081.
[8]. Madhusduhana Rao. D. and Srinivasa Rao. G., A Study on Ternary Semirings, International Journal of Mathematical Archive-
5(12), Dec.-2014, 24-30.
[9]. MadhusudhanaRao. D. and SrinivasaRao. G., Special Elements of a Ternary Semirings, International Journal of Engineering
Research and Applications, Vol. IV, Issue 1(Version-5), November 201IV, pp. 123-130.
[10]. MadhusudhanaRao. D. and SrinivasaRao. G.,Structure of Certain Ideals in Ternary Semirings-International Journal of Innovative
Science and Modern Engineering(IJISME)-Volume-3, Issue-2, January, 2015.
7. P-Pseudo Symmetric Ideals in Ternary Semiring
DOI: 10.9790/5728-1115105111 www.iosrjournals.org 111 | Page
[11]. Madhusudhana Rao. D. and Srinivasa Rao. G. Structure of P-Prime and P-Semiprime Ideals in Ternary Semirings- accepted for
publication in International Journal of Innovation in Science and Mathematics(IJISM) Vol 3, Issue 2, February 2015.
[12]. Madhusudhana Rao. D. and Srinivasa Rao. G. Prime Radicals and Completely Prime Radicals in Ternary Semirings-International
Journal of Basic Science and Applied Computing (IJBSAC) Volume-1, Issue 4, February 2015.
[13]. Sarala. Y, Anjaneyulu. A and Madhusudhana Rao. D. Pseudo Symmetric Ideals in Ternary Semigroups-International Refereed
Journal of Engineering and Science, Volume-1, Issue-4, (December 2012), pp.33-43.
Author’s Biography
Dr D. MadhusudhanaRao completed his M.Sc. from Osmania University, Hyderabad, Telangana,
India. M. Phil. from M. K. University, Madurai, Tamil Nadu, India. Ph. D. from
AcharyaNagarjuna University, Andhra Pradesh, India. He joined as Lecturer in Mathematics, in
the department of Mathematics, VSR & NVR College, Tenali, A. P. India in the year 1997, after
that he promoted as Head, Department of Mathematics, VSR & NVR College, Tenali. He helped
more than 5 Ph.D’s and at present he guided 5 Ph. D. Scalars and 3 M. Phil., Scalars in the
department of Mathematics, AcharyaNagarjuna University, Nagarjuna Nagar, Guntur, A. P.A major part of his
research work has been devoted to the use of semigroups, Gamma semigroups, duo gamma semigroups,
partially ordered gamma semigroups and ternary semigroups, Gamma semirings and ternary semirings, Near
rings ect. He acting as peer review member to the “British Journal of Mathematics & Computer Science”.
He published more than 40 research papers in different International Journals in the last three academic years.
G. Srinivasarao: He is working as an Assistant Professor in the Department of Applied Sciences &
Humanities, Tirumala Engineering College. He completed his M.Phil. inMadhuraiKamaraj
University . He is pursuing Ph.D. under the guidance of Dr.D.Madhusudanarao in
AcharyaNagarjuna University. He published more than 7 research papers in popular international
Journals to his credit. His area of interests are ternary semirings, ordered ternary semirings,
semirings and topology. Presently he is working on Ternary semirings
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