• Kyungpook Mathematical Journal
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• v.48 no.4
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• pp.585-591
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• 2008

In 1981, Sen [4] have introduced the concept of $\Gamma$-semigroups. We have known that $\Gamma$-semigroups are a generalization of semigroups. In this paper, we introduce the concepts of the extensions of s-prime ideals, prime ideals, s-semiprime ideals and semiprime ideals in $\Gamma$-semigroups and characterize the relationship between the extensions of ideals and some congruences in $\Gamma$-semigroups.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.425-436
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• 2022

In this paper we introduce the notion of 2-absorbing primary ideals of a commutative semigroup. We establish the relations between 2-absorbing primary ideals and prime, maximal, semiprimary and 2-absorbing ideals. We obtain various characterization theorems for commutative semigroups in which 2-absorbing primary ideals are prime, maximal, semiprimary and 2-absorbing ideals. We also study some other important properties of 2-absorbing primary ideals of a commutative semigroup.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.51-62
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• 2015

Pseudo-ŁBCK-algebras are commutative pseudo-BCK-algebras with relative cancellation property. In the paper, we introduce fuzzy prime ideals in pseudo-ŁBCK-algebras and investigate some of their properties. We also give various characterizations of prime ideals and fuzzy prime ideals. Moreover, we present conditions for a pseudo-ŁBCKalgebra to be a pseudo-ŁBCK-chain.

• Kyungpook Mathematical Journal
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• v.54 no.4
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• pp.629-638
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• 2014

The concept of prime and weakly prime ideal in semigroups has been introduced by G. Szasz [4]. In this paper, we define the involution in po-${\Gamma}$-semigroups, then we extend some results on prime, semiprime and weakly prime ideals to the involution po-${\Gamma}$-semigroup S. Also, we characterize intra-regular involution po-${\Gamma}$-semigroups. We establish that in the involution po-${\Gamma}$-semigroup S such that the involution preserves the order, an ideal of S is prime if and only if it is both weakly prime and semiprime and if S is commutative, then the prime and weakly prime ideals of S coincide. Finally, we prove that if S is a po-${\Gamma}$-semigroup with order preserving involution, then the ideals of S are prime if and only if S is intra-regular.

• Korean Journal of Mathematics
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• v.13 no.1
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• pp.13-18
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• 2005

Recently, Kehayopulu and Tsingelis studied for prime ideals of groupoids-ordered groupoids. In this paper, we give some results on prime left(right) ideals of groupoid-ordered groupoid. These results are generalizations of their results.

• Communications of the Korean Mathematical Society
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• v.13 no.2
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• pp.251-256
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• 1998

We introduce the concept of weakly prime ideals in po-$\Gamma$-semigroup and give some characterizations of weakly prime ideals.

• Honam Mathematical Journal
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• v.27 no.3
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• pp.333-341
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• 2005

A characterization of prime ideals is discussed. A relation between prime ideals and ideals of the form $A_w^{\wedge}$ is given. The prime ideal theorem is established. The notion of annihilators is introduced, and basic properties are investigated.

• Kyungpook Mathematical Journal
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• v.54 no.1
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• pp.123-129
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• 2014

In this paper the concepts of pure ideals, weakly pure ideals and purely prime ideals in ordered semigroups are introduced. We obtain some characterizations of pure ideals and prove that the set of all pure prime ideals is topologized.

• Journal of applied mathematics & informatics
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• v.34 no.5_6
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• pp.405-419
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• 2016

The notion of Γ-semiring was introduced by M. Murali Krishna Rao [8] as a generalization of Γ-ring as well as of semiring. In this paper fuzzy k-ideals, k-fuzzy ideals and fuzzy-2-prime ideals in Γ-semirings have been introduced and study the properties related to them. Let μ be a fuzzy k-ideal of Γ-semiring M with |Im(μ)| = 2 and μ(0) = 1. Then we establish that M_μ is a 2-prime ideal of Γ-semiring M if and only if μ is a fuzzy prime ideal of Γ-semiring M.

• Communications of the Korean Mathematical Society
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• v.23 no.1
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• pp.19-27
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• 2008

The motivation mainly comes from the conditions of the (ordered) ideals to be prime or semiprime that are of importance and interest in (ordered) semigroups and in (ordered) $\Gamma$-semigroups. In 1981, Sen [8] has introduced the concept of the $\Gamma$-semigroups. We can see that any semigroup can be considered as a $\Gamma$-semigroup. The concept of ordered ideal extensions in ordered $\Gamma$-semigroups was introduced in 2007 by Siripitukdet and Iampan [12]. Our purpose in this paper is to introduce the concepts of the ordered n-prime ideals and the ordered n-semiprime ideals in ordered $\Gamma$-semigroups and to characterize the relationship between the ordered n-prime ideals and the ordered ideal extensions in ordered $\Gamma$-semigroups.