The least regular order with respect to a regular congruence on ordered Γ-semigroups

Abstract

The motivation mainly comes from the conditions of congruences to be regular that are of importance and interest in ordered semigroups. In 1981, Sen has introduced the concept of the Γ-semigroups. We can see that any semigroup can be considered as a Γ-semigroup. In this paper, we introduce and characterize the concept of the regular congruences on ordered Γ-semigroups and prove the following statements on an ordered Γ-semigroup M :(1) Every ordered semilattice congruences is a regular congruence.(2) There exists the least regular order on the Γ-semigroup M/ρ with respect to a regular congruence ρ on M .(3) The regular congruences are not ordered semilattice congruences in general. PreliminariesRecall that a semilattice congruence N on an ordered semigroup S is an ordered semilattice congruence on S. Thus the S/N endowed with the multiplication (x) N · (y) N = (xy) N is still a semigroup. If we define (x) N (y) N if and only if (x) N = (xy) N , then it can be easily seen that (S/N ; ·, ) is an ordered semigroup. We now call this semigroup (S/N ; ·, ) the natural ordered semigroup induced by the ordered semilattice congruence N on S. In 1997, Xie and Wu [1] proved that the semilattice congruence N on an ordered semigroup S is the least regular semilattice congruence. In 2004, Dutta and Adhikari [2] introduced the concepts of ordered Γ-semigroups. In 2004, Xu and Ma [3] showed that there exists an orderpreserving bijection between the set of all prime ideals of the ordered semigroup (S; ·, ≤) and the set of all prime ideals of (S/N ; ·, ). Moreover, they gave some necessary and sufficient conditions for the natural ordered semigroup (S/N ; ·, ) to be a chain. In 2006, Siripitukdet and Iampan [4] characterized the relationship among the (ordered) filters, (ordered) s-prime ideals and (ordered) semilattice congruences in ordered Γ-semigroups and gave some characterizations of semilattice congruence and ordered semilattice congruence on ordered Γ-semigroups. They proved that

THE FIRST ISOMORPHISM THEOREM FOR (CO-ORDERED) $\Gamma$-SEMIGROUPS WITH APARTNESS

“…Γ-semigroup with apartness. To explain the notions and notations used in this article, which but not previously described, we instruct a reader to look at the articles [11,12,25,26,27,28]. Here we will introduce some specific substructures of this semigroups that appear only in the Bish version.…”

“…Also, we will find and analyze some doubles of substructures of these semigroups. Our investigation the concept of Γ-semigroups with appartness consists of the observation of specificities that arise by placing the classically defined algebraic structure of Γ-semigroups ( [5,9,10,11,12,25,26,27,28]) into a different logical environment and using specific Bishop's constructive algebra tools. Important for this analysis were the articles [5,10] in which the isomorphism theorems in such semigroups are treated while in the papers [11,12,28] the properties of ordered Γ-semigroups are observed.…”

On inf -Hesitant Fuzzy Γ -Ideals of Γ -Semigroups

“…In 1981, Sen [17] introduced the concept and notion of the Γ-semigroup as a generalization of the plain semigroup and ternary semigroup. Many classical notions and results of (ternary) semigroups have been extended and generalized to Γ-semigroups, by many mathematicians, for instance, Siripitukdet and Iampan [18,19], Siripitukdet and Julatha [20], Dutta and Adhikari [21,22], Saha and Sen [23][24][25], Hila [26,27], and Chinram [28,29]. Simuen, Iampan, Chinram, Sardar, Majumder, Dutta, and Davvaz [30][31][32][33][34][35] studied theory of Γ-semigroups via fuzzy subsets.…”