Special Types of Ternary Semigroups V . Jyothi (original) (raw)

IDEALS IN TERNARY SEMIGROUPS

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.

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.

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

x T\U or T is a simple ternarysemigroup. Definition 2.1: Let T ≠ . Then T is called a ternarysemigroup if being existence a mapping from T T T to T which maps (pqr) [ pqr] satisfying the condition :[(pqr) st] = [ p(qrs)t] = [pq(rst)] for all p, q, r, s, t T. Definition 2.2: An idempotent component e T is said to be left (or lateral or right) identity of the if eaa = a(or aea = a or aae = a) for all a T. Left (or lateral or right) identity may not be unique. But if e is an identity (i.e. e plays the role of left lateral and right identity simultaneously) then e is unique. V. Jyothi et al.

PRIMARY IDEALS IN TERNARY SEMIGROUPS

In this paper, the terms left primary ideal, lateral primary ideal, right primary ideal, primary ideal, left primary ternary semigroup, lateral primary ternary semigroup , right primary ternary semigroup, primary ternary semigroup are introduced. It is proved that A be an ideal in a ternary semigroup T and if X, Y, Z are three ideals of T such that 1)

Prime Radicals in Ternary Semigroups

In this paper the terms completely prime ideal, prime ideal, m-system. globally idempotent , semi simple elements of a ternary semigroup are Introduced. It is proved that an ideal A of a ternary semigroup T is completely prime if and only if T\A is either sub semigroup of T or empty. It is proved that if T is a globally idempotent ternary semigroup then every maximal ideal of T is a prime ideal of T. In this paper the terms completely semiprime ideal, semiprime ideal, n-system, d-system and i-system are introduced. It is proved that the non-empty intersection of any family of a completely prime ideal and prime ideal of a ternary semigroup T is a

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

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.