• Korean Journal of Mathematics
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• 제23권3호
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• pp.357-370
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• 2015

In this paper, we introduce the concept of (m, n)-ideals in a non-associative ordered structure, which is called an ordered Abel-Grassmann's groupoid, by generalizing the concept of (m, n)-ideals in an ordered semigroup [14]. We also study the (m, n)-regular class of an ordered AG-groupoid in terms of (m, n)-ideals.

• 호남수학학술지
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• 제35권4호
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• pp.583-606
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• 2013

We generalize the idea of (${\in}$, ${\in}{\vee}q_k$)-fuzzy ordered semi-group and give the concept of (${\in}$, ${\in}{\vee}q_k$)-fuzzy ordered $\mathcal{LA}$-semigroup. We show that (${\in}$, ${\in}{\vee}q_k$)-fuzzy left (right, two-sided) ideals, (${\in}$, ${\in}{\vee}q_k$)-fuzzy (generalized) bi-ideals, (${\in}$, ${\in}{\vee}q_k$)-fuzzy interior ideals and (${\in}$, ${\in}{\vee}q_k$)-fuzzy (1, 2)-ideals need not to be coincide in an ordered $\mathcal{LA}$-semigroup but on the other hand, we prove that all these (${\in}$, ${\in}{\vee}q_k$)-fuzzy ideals coincide in a left regular class of an ordered $\mathcal{LA}$-semigroup. Further we investigate some useful conditions for an ordered $\mathcal{LA}$-semigroup to become a left regular ordered $\mathcal{LA}$-semigroup and characterize a left regular ordered $\mathcal{LA}$-semigroup in terms of (${\in}$, ${\in}{\vee}q_k$)-fuzzy one-sided ideals. Finally we connect an ideal theory with an (${\in}$, ${\in}{\vee}q_k$)-fuzzy ideal theory by using the notions of duo and (${\in}{\vee}q_k$)-fuzzy duo.

• 충청수학회지
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• 제22권2호
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• pp.235-243
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• 2009

In this paper, we introduce the notion of intuitionistic fuzzy semiprimality in an ordered semigroup, which is an extension of fuzzy semiprimality and investigate some properties of intuitionistic fuzzification of the concept of several ideals.

• Korean Journal of Mathematics
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• 제6권2호
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• pp.149-154
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• 1998

We consider the ordered ${\Gamma}$-semigroups in which $x{\gamma}x(x{\in}M,{\gamma}{\in}{\Gamma})$ are left elements. We show that this $po-{\Gamma}$-semigroup is left regular if and only if M is a union of left simple sub-${\Gamma}$-semigroups of M.

• 대한수학회논문집
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• 제11권2호
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• pp.315-321
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• 1996

Recently, N. Kehayopulu [4] introduced the concepts of weakly prime ideals of ordered semigroups. In this paper, we define the concepts of left(right) weakly prime and left(right) semiregular. Also we investigate the properties of them.

• 대한수학회논문집
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• 제13권1호
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• pp.1-6
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• 1998

The paper refers to ordered semigroups in which $x^2 (x \in S)$ are left ideal elements. We mainly show that this $po$-semigroup is left regular if and only if S is a union of left simple subsemigroups of S.

• 대한수학회지
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• 제46권4호
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• pp.775-784
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• 2009

In 1932, Lehmer [4] gave the definition of a ternary semigroup. We can see that any semigroup can be reduced to a ternary semigroup. In this paper, we give some auxiliary results which are also necessary for our considerations and characterize the relationship between the (0-)minimal and maximal ordered lateral ideals and the lateral simple and lateral 0-simple ordered ternary semigroups analogous to the characterizations of minimal and maximal left ideals in ordered semigroups considered by Cao and Xu [2].

• Kyungpook Mathematical Journal
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• 제57권4호
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• pp.545-558
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• 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.

• Journal of applied mathematics & informatics
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• 제27권5_6호
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• pp.1447-1457
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• 2009

In this paper we define intuitionistic fuzzy interior ideals in ordered semigroups. We prove that in regular(resp. intra-regular and semisimple) ordered semigroups the concepts of intuitionistic fuzzy interior ideals and intuitionistic fuzzy ideals coincide. We prove that an ordered semi group is intuitionistic fuzzy simple if and only if every intutionistic fuzzy interior ideal is a constant function. We characterize intra-regular ordered semi groups in terms of interior (resp. intuitionistic fuzzy interior) ideals.

• 호남수학학술지
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• 제36권3호
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• pp.569-596
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• 2014

In several applied disciplines like control engineering, computer sciences, error-correcting codes and fuzzy automata theory, the use of fuzzied algebraic structures especially ordered semi-groups and their fuzzy subsystems play a remarkable role. In this paper, we introduce the notion of (${\in},{\in}{\vee}\bar{q}_k$)-fuzzy subsystems of ordered semigroups namely (${\in},{\in}{\vee}\bar{q}_k$)-fuzzy generalized bi-ideals of ordered semigroups. The important milestone of the present paper is to link ordinary generalized bi-ideals and (${\in},{\in}{\vee}\bar{q}_k$)-fuzzy generalized bi-ideals. Moreover, different classes of ordered semi-groups such as regular and left weakly regular ordered semigroups are characterized by the properties of this new notion. Finally, the upper part of a (${\in},{\in}{\vee}\bar{q}_k$)-fuzzy generalized bi-ideal is defined and some characterizations are discussed.