• 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].

• 대한수학회보
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• 제51권4호
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• pp.1217-1227
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• 2014

The intersection of all two-sided ideals of an ordered semigroup, if it is non-empty, is called the kernel of the ordered semigroup. A left ideal L of an ordered semigroup ($S,{\cdot},{\leq}$) having a kernel I is said to be simple if I is properly contained in L and for any left ideal L' of ($S,{\cdot},{\leq}$), I is properly contained in L' and L' is contained in L imply L' = L. The notions of simple right and two-sided ideals are defined similarly. In this paper, the author characterize when an ordered semigroup having a kernel is the class sum of its simple left, right and two-sided ideals. Further, the structure of simple two-sided ideals will be discussed.

• 대한수학회논문집
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• 제28권2호
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• pp.225-229
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• 2013

According to the paper in [3], the kernel of an ordered semigroup S is a completely regular subsemigroup of S. In this note we show that the kernel of an ordered semigroup S is not a completely regular subsemigroup of S, in general.

• Journal of applied mathematics & informatics
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• 제28권1_2호
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• pp.311-324
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• 2010

We prove that a regular ordered semigroup S is left simple if and only if every intuitionistic fuzzy left ideal of S is a constant function. We also show that an ordered semigroup S is left (resp. right) regular if and only if for every intuitionistic fuzzy left(resp. right) ideal A = <$\mu_A$, $\gamma_A$> of S we have $\mu_A(a)\;=\;\mu_A(a^2)$, $\gamma_A(a)\;=\;\gamma_A(a^2)$ for every $a\;{\in}\;S$. Further, we characterize some semilattices of ordered semigroups in terms of intuitionistic fuzzy left(resp. right) ideals. In this respect, we prove that an ordered semigroup S is a semilattice of left (resp. right) simple semigroups if and only if for every intuitionistic fuzzy left (resp. right) ideal A = <$\mu_A$, $\gamma_A$> of S we have $\mu_A(a)\;=\;\mu_A(a^2)$, $\gamma_A(a)\;=\;\gamma_A(a^2)$ and $\mu_A(ab)\;=\;\mu_A(ba)$, $\gamma_A(ab)\;=\;\gamma_A(ba)$ for all a, $b\;{\in}\;S$.

• 한국지능시스템학회논문지
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• 제17권1호
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• pp.118-122
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• 2007

In this paper, we consider the intuitionistic fuzzification of the notion of a interior ideal in ordered semigroup S, and investigate some properties of such ideals. In terms of intuitionistic fuzzy set, characterizations of intuitionistic fuzzy interior ideals in ordered semigroups are discussed. Using a collection of interior ideals with additional conditions, an intuitionistic fuzzy interiror ideal is constructed. Natural equivalence relations on the set of all intuitionistic fuzzy interior ideals of an ordered semigroup are investigated. We also give a characterization of a intuitionistic fuzzy simple semigroup in terms of intuitionistic fuzzy interior ideals.

• 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.

• 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.

• 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.

• 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.

• Journal of applied mathematics & informatics
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• 제28권5_6호
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• pp.1217-1225
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• 2010

In this paper we define fuzzy interior $\Gamma$-ideals in ordered $\Gamma$-semigroups. We prove that in regular(resp. intra-regular) ordered $\Gamma$-semigroups the concepts of fuzzy interior $\Gamma$-ideals and fuzzy $\Gamma$-ideals coincide. We prove that an ordered $\Gamma$-semigroup is fuzzy simple if and only if every fuzzy interior $\Gamma$-ideal is a constant function. We characterize intra-regular ordered $\Gamma$-semigroups in terms of interior (resp. fuzzy interior) $\Gamma$-ideals.