PRIMARY IDEALS IN TERNARY SEMIGROUPS (original) (raw)
In this paper the terms ideal, trivial ideal, proper ideal, maximal ideal are introduced. It is proved that the union and intersection of any family of ideals of ternary semigroup T is an ideal of T. It is also proved that union of all proper ideals of ternary semigroup T is the unique maximal ideal of T. The terms ideal of ternary semigroup T generated by A, principal ideal generated by an element are introduced. It is proved that the ideal of a ternary semigroup T generated by a non-empty subset A is the intersection of all ideals of T containing A. It is also proved that T is a ternary semigroup and a T then J(a) = a aTT TTa TaT TTaTT . The terms, simple ternary semigroup, globally idempotent ideal are introduced. In any ternary semigroup T, principal ideals of T form a chain and ideals of T form a chain are equivalent. It is proved that a ternary semigroup T is simple ternary semigroup if and only if TTaTT = T for all a T. It is also proved that if T is a globally idempotent ternary semigroup having maximal ideals then T contains semisimple elements.
Pseudo Symmetric Ideals In Ternary Semigroups
In this paper the terms pseudo symmetric ideals, pseudo symmetric ternary semigroups, semipseudo symmetric ideals and semipseudo symmetric ternary semigroups. It is proved that for any pseudo symmetric ideal A in a ternary semigroup T, for any natural number n, a 1 a 2 … a n-1 a n ∈ A if and only if < a 1 > < a 2 > ………… < a n > ⊆ A. It is proved that every completely semiprime ideal of a ternary semigroup is a pseudo symmetric ideal. Further it is proved that an ideal A of a ternary semigroup is (1) completely prime iff A is prime and pseudo symmetric, (2) completely semiprime iff A is semiprime and pseudo symmetric. It is also proved that every prime ideal P minimal relative to containing a pseudo symmetric ideal A in a ternary semigroup T is completely prime and hence every prime ideal P minimal relative to containing a completely semiprime ideal A in a ternary semigroup T is completely prime. It is proved that every pseudo commutative ternary semigroup, ternary semigroup in which every element is a mid unit, are pseudo symmetric ternary semigroups. It is proved that every pseudo symmetric ideal of a ternary semigroup is a semipseudo symmetric ideal. It is also proved that every semiprime ideal P minimal relative to containing a semipseudo symmetric ideal A of a ternary semigroup is completely semiprime. If A is a semipseudo symmetric ideal of a ternary semigroup T, then A 1 = the intersection of all completely prime ideals of T containing A, (2) 1 A = the intersection of all minimal completely prime ideals of T containing A, (3) 1 A = the minimal completely semiprime ideal of T relative to containing A, (4) A 2 = {x ∈ T : x n ∈ A for some odd natural number n},(5) A 3 = the intersection of all prime ideals of T containing A, (6) 3 A = the intersection of all minimal prime ideals of T containing A, (7) 3
A note on Quasi and Bi-ideals in Ternary semigroups
International Journal of Mathematics and Mathematical Sciences, 1995
In this paper we have studied the properties of Quasi-ideals and Bi-ideals in ternary semi groups. We prove that every quasi-ideal is a bi-ideal inTbut the converse is not true in general by giving several example in different context.
On ideals in regular ternary semigroups
Discussiones Mathematicae - General Algebra and Applications, 2008
In this paper we study some interesting properties of regular ternary semigroups, completely regular ternary semigroups, intra-regular ternary semigroups and characterize them by using various ideals of ternary semigroups.
Special Types of Ternary Semigroups V . Jyothi
2014
The main goal of this paper is to initiate the notions of U-ternarysemigroup and V-ternary semigroup in the class of orbitary ternarysemigroups. We study prime ideals and maximal ideals in a Uternarysemigroup and characterize V-ternary semigroup. It is proved that if T is a globally idempotent ternarysemigroups with maximal ideal, then either T is a V-ternarysemigroup or T has a unique maximal ideal which is prime. Finally we proved that a ternarysemigroup T is a V-ternarysemigroup if and only if T has atleast one proper prime ideal and if { } is the family of all proper prime ideals, then < x > =T for x T\U or T is a simple ternarysemigroup.
On Generalised Quasi-ideals in Ordered Ternary Semigroups
2017
In this paper, we introduce generalised quasi-ideals in ordered ternary semigroups. Also, we define ordered m-right ideals, ordered (p, q)-lateral ideals and ordered n-left ideals in ordered ternary semigroups and studied the relation between them. Some intersection properties of ordered (m, (p, q), n)-quasi ideals are examined. We also characterize these notions in terms of minimal ordered (m, (p, q), n)-quasi-ideals in ordered ternary semigroups. Moreover, m-right simple, (p, q)-lateral simple, n-left simple, and (m, (p, q), n)-quasi simple ordered ternary semigroups are defined and some properties of them are studied.
Generalised Bi-ideals in ordered ternary semigroups
2017
The aim of this paper is to introduce a new concept of an ordered (m; (p; q); n)-bi ideal of an ordered tenary semigroup. Some classical results in ordered ternary semigroups are given. We also consider the minimal ordered (m; (p; q); n)-bi ideals in ordered ternary semigroups. In particular, the (m; (p; q); n)-bi simple ordered ternary semigroups are dened and some of their properties are explored. As a conse- quence, we will show that the regular ordered ternary semigroups can be characterized by using various generalised ideals.
Chained Commutative Ternary Semigroups
In this paper, the terms chained ternary semigroup, cancellable clement , cancellative ternary semigroup, A-regular element, π-regular element, πinvertible 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 n1
PRIMARY IDEALS IN QUASI-COMMUTATIVE TERNARY SEMIGEOUPS
In this paper we study the structure of cancellative quasi-commutative primary ternary semigroups. In fact we prove that if T is a cancellative quasi-commutative ternary semigroup, then (1) S is a primary ternary semigroup proper prime ideals in T are maximal and (3) semiprimary ideals in T are primary, are equivalent. Mathematics Subject Classification (2000): 20M12, 20M17.
On 3-prime and quasi 3-primary ideals of ternary semirings
Discussiones Mathematicae. General Algebra and Applications, 2024
The purpose of this paper is to introduce the concept of 3-prime ideal as a generalization of prime ideal. Further, we generalize the concepts of 3-prime ideal and primary ideal, namely as quasi 3-primary ideal in a commutative ternary semiring with zero. The relationship among prime ideal, 3-prime ideal, primary ideal, quasi primary and quasi 3-primary ideal are investigated. Various results and examples concerning 3-prime ideals and quasi 3-primary ideals are given. Analogous theorems to the primary avoidance theorem for quasi 3-primary ideals are also studied.
Prime Ideals in Ternary Semigroups
Asian-European Journal of Mathematics, 2009
In this paper we define prime, semiprime and irreducible ideals in ternary semigroups. We also define semisimple ternary semigroups and prove that a ternary semigroup is semisimple if and only if each of its ideals is semiprime.
A STUDY ON d-SYSTEM, m-SYSTEM AND n-SYSTEM IN TERNARY SEMIGROUPS
Qualitative In this paper the terms d-system, m-system, n-system, U-ternary semigroup are introduced. It is proved that an ideal A of a ternary semigroup T is a prime ideal of T if and only if T\A is an m-system of T or empty. It is proved that an ideal A of a ternary semigroup T is completely semiprime if and only if T\A is a d-system of T or empty. It is proved that every msystem in a ternary semigroup T is an n-system. Further it is proved that an ideal Q of a ternary semigroup T is a semiprime ideal if and only if T\Q is an n-system of T (or) empty. It is proved that if N is an n-system in a ternary semigroup T and a N, then there exist an m-system M in T such that a M and M N. It is proved that a ternary semigroup T is U-ternary semigroup if
Characterization of ternary semigroups in terms of (∈,∈∨ qk) ideals
In this paper, we present the concepts of (∈, ∈ ∨q k)-fuzzy ideals in ternary semigroups, which is a generalization of the (∈, ∈ ∨q) fuzzy ideals of a ternary semigroups. In this regard, we define (∈, ∈ ∨q k)-fuzzy left (right, lateral) ideals, (∈, ∈ ∨q k)-fuzzy quasi-ideals and (∈, ∈ ∨q k)fuzzy bi-ideals and prove some basic results using these definitions. Special concentration is paid to (∈, ∈ ∨q k)-fuzzy left (right, lateral) ideals, (∈, ∈ ∨q k)-fuzzy quasi-ideals and (∈, ∈ ∨q k)-fuzzy bi-ideals. Furthermore, we characterize regular ternary semigroups in terms of these notions.
IDEALS IN REGULAR Po -TERNARY SEMIGROUPS
Ideals play an important role in the Algebraic structures like rings, semigroups and semirings. In this paper some generalizations of ideals in regular partially ordered (po) gamma ternary semigroups are studied. It is proved that " every ideal of a regular po-gamma-ternary semigroup T is semiprime " and some equivalent conditions.
On irreducible pseudo symmetric ideals of a partially ordered ternary semigroup
Quasigroups and Related Systems, 2022
In this paper, the concepts of irreducible and strongly irreducible pseudo symmetric ideals in a partially ordered ternary semigroup are introduced. We also studied some interesting properties of irreducible and strongly irreducible pseudo symmetric ideals of a partially ordered ternary semigroup and prove that the space of strongly irreducible pseudo symmetric ideals of a partially ordered ternary semigroup is topologized.
Radicals of generalized prime ideals in ternary semigroups
Discussiones Mathematicae - General Algebra and Applications
In this paper, the concepts of f-prime ideals and f-semiprime ideals on a ternary semigroup are considered as a generalization of pseudo prime ideals and pseudo semiprime ideals, respectively. Then such ideals introduced are used to describe left (respectively, right) f-primary ideals on a ternary semigroup.