On right chain ordered semigroups

Changphas, Thawhat; Luangchaisri, Panuwat; Mazurek, Ryszard

doi:10.1007/s00233-017-9896-z

Cited by 7 publications

(27 citation statements)

References 9 publications

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“…Then, this concept was moved to semigroups by Ferrero, Mazurek, and Sant'Ana [2] . Finally, it was extended to ordered semigroups by Luangchaisri, Changphas, and Mazurek in [3].…”

Section: Introductionmentioning

confidence: 99%

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Weakly Semiprime Segments in Ordered Semigroups

Luangchaisri

Changphas

2019

Mathematics

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“…Then, this concept was moved to semigroups by Ferrero, Mazurek, and Sant'Ana [2] . Finally, it was extended to ordered semigroups by Luangchaisri, Changphas, and Mazurek in [3].…”

Section: Introductionmentioning

confidence: 99%

“…[3] An ideal P of an ordered semigroup (S, ·, ≤) is prime if and only if P is both semiprime and weakly prime.…”

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confidence: 99%

Weakly Semiprime Segments in Ordered Semigroups

Luangchaisri

Changphas

2019

Mathematics

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“…Clifford uses the term "semiprime ideal" and Petrich the term "completely semiprime ideal (subset)". For ordered semigroups I adopted the terminology due to Clifford; the authors in [2] the terminology by Petrich. Since in the present paper we refer to [2], for the sake of completeness, in particular for this paper, we will use the terms prime, semiprime, completely prime, completely semiprime.…”

Section: Introduction and Prerequisitesmentioning

confidence: 99%

“…For ordered semigroups I adopted the terminology due to Clifford; the authors in [2] the terminology by Petrich. Since in the present paper we refer to [2], for the sake of completeness, in particular for this paper, we will use the terms prime, semiprime, completely prime, completely semiprime. For an ordered semigroup S the zero of S, denoted by 0, is an element of S such that 0x = x0 = 0 and 0 ≤ x for every x ∈ S [1,3].…”

Section: Introduction and Prerequisitesmentioning

confidence: 99%

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On Hoehnke Ideal in Ordered Semigroups

Kehayopulu

2018

Eur. J. Pure Appl. Math.

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For a proper subset $A$ of an ordered semigroup $S$, we denote by $H_A(S)$ the subset of $S$ defined by $H_A(S):=\{h\in S \mbox { such that if } s\in S\backslash A, \mbox { then } s\notin (shS]\}$. We prove, among others, that if $A$ is a right ideal of $S$ and the set $H_A(S)$ is nonempty, then $H_A(S)$ is an ideal of $S$; in particular it is a semiprime ideal of $S$. Moreover, if $A$ is an ideal of $S$, then $A\subseteq H_A(S)$. Finally, we prove that if $A$ and $I$ are right ideals of $S$, then $I\subseteq H_A(S)$ if and only if $s\notin (sI]$ for every $s\in S\backslash A$. We give some examples that illustrate our results. Our results generalize the Theorem 2.4 in Semigroup Forum 96 (2018), 523--535.

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On completely regular and Clifford ordered semigroups

Bhuniya

Hansda

2020

Afr. Mat.

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Lee and Kwon [12] defined an ordered semigroup S to be completely regular if a ∈ (a 2 Sa 2 ] for every a ∈ S. We characterize every completely regular ordered semigroup as a union of t-simple subsemigroups, and every Clifford ordered semigroup as a complete semilattice of t-simple subsemigroups. Green's Theorem for the completely regular ordered semigroups has been established. In an ordered semigroup S, we call an element e an ordered idempotent if it satisfies e ≤ e 2 . Different characterizations of the regular, completely regular and Clifford ordered semigroups are done by their ordered idempotents. Thus a foundation for the completely regular ordered semigroups and Clifford ordered semigroups has been developed. * correspondingauthor T. Saito studied systematically the influence of order on regular, idempotent, inverse, and completely regular semigroups [14] -[17], whereas Kehayopulu, Tsingelis, Cao and many others characterized regularity, complete regularity, etc. on ordered semigroups [1] -[3], [7]- [12]. Success attained by the school characterizing regularity on ordered semigroups are either in the semilattice and complete semilattice decompositions into different types of simple components, viz. left, t-, σ, λ-simple etc. or in its ideal theory.Complete regularity on ordered semigeoups was introduced by Lee and Kwon [12]. Here we give their complete semilattice decomposition and express them as a union of t-simple ordered semigroups. This supports the observation of Cao [3] that the t-simple ordered semigroups plays the same role in the theory of ordered semigroups as groups in the theory of semigroups without order. Then it follows that the semigroups which are semilattices of t-simple ordered semigroups are the analogue of Clifford semigroups. Though it is not under the name Clifford ordered semigroups, but such ordered semigroups have been studied extensively by Cao [2] and Kehayopulu [10], specially complete semilattice decomposition of such semigroups. We generalize such ordered semigroups into left Clifford ordered semigroups.Another successful part of this paper is identification of the ordered idempotent elements in an ordered semigroup and exploration of their behavior in both completely regular and Clifford ordered semigroups. Also an extensive study has been done on the idempotent ordered semigroups. Complete semilattice decomposition of these semigroups automatically suggests the looks of rectangular idempotent semigroups and in this way we arrive to many other important classes of idempotent ordered semigroups.The presentation of the article is as follows. This section is followed by preliminaries. In Section 3, basic properties of the t−simple ordered semigroups which we call here group like ordered semigroups have been studied.Completely regular ordered semigroups have been characterized in Section 4. Section 5 is devoted to the the Clifford ordered semigroups and their generalizations. PreliminariesAn ordered semigroup is a partially ordered set (S, ≤), and at the same time a semigrou...

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On right chain ordered semigroups

Cited by 7 publications

References 9 publications

Weakly Semiprime Segments in Ordered Semigroups

Weakly Semiprime Segments in Ordered Semigroups

On Hoehnke Ideal in Ordered Semigroups

On completely regular and Clifford ordered semigroups

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