Prime Ideals in Ternary Semigroups (original) (raw)
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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
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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.
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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.
Special Types of Ternary Semigroups V . Jyothi
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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.
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.
Radicals of generalized prime ideals in ternary semigroups
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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.
PRIMARY IDEALS IN TERNARY SEMIGROUPS
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Completely Prime Ideals of Semigroups
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In this paper we discuss a very useful tool in the study of semigroup theory, called completely prime ideals. We will study characterization of semilattice congruences by means of completely prime ideals. We will follow the procedure of finding all subdirectly irreducible semilattices, then construct all congruences induced by homomorphism onto these, and finally take arbitrary intersections of congruences constructed. The usual algebraic books and papers studied semilattice congruence of semigroups by means of strongly prime ideals. We give some necessary and sufficient conditions about completely semiprime ideals, semilattice congruence and filters, which provide clearer informations about semigroups structure and new algebraic approach of studying them. Also we study the relationship between-simple semigroup and completely prime ideals. So the purpose of this paper is twofold: to characterize an arbitrary nonempty intersection of completely prime ideals and characterize subsets which are congruence classes of some semilattice congruence.