• Title/Summary/Keyword: $\Gamma$-semigroup

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EPIMORPHISMS, DOMINIONS FOR GAMMA SEMIGROUPS AND PARTIALLY ORDERED GAMMA SEMIGROUPS

  • PHOOL MIYAN;SELESHI DEMIE;GEZEHEGN TEREFE
    • Journal of applied mathematics & informatics
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    • v.41 no.4
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    • pp.707-722
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    • 2023
  • The purpose of this paper is to obtain the commutativity of a gamma dominion for a commutative gamma semigroup by using Isbell zigzag theorem for gamma semigroup and we prove some gamma semigroup identities are preserved under epimorphism. Moreover, we extend epimorphism, dominion and Isbell zigzag theorem for partially ordered semigroup to partially ordered gamma semigroup.

E-Inversive Γ-Semigroups

  • Sen, Mridul Kanti;Chattopadhyay, Sumanta
    • Kyungpook Mathematical Journal
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    • v.49 no.3
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    • pp.457-471
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    • 2009
  • Let S = {a, b, c, ...} and ${\Gamma}$ = {${\alpha}$, ${\beta}$, ${\gamma}$, ...} be two nonempty sets. S is called a ${\Gamma}$-semigroup if $a{\alpha}b{\in}S$, for all ${\alpha}{\in}{\Gamma}$ and a, b ${\in}$ S and $(a{\alpha}b){\beta}c=a{\alpha}(b{\beta}c)$, for all a, b, c ${\in}$ S and for all ${\alpha}$, ${\beta}$ ${\in}$ ${\Gamma}$. An element $e{\in}S$ is said to be an ${\alpha}$-idempotent for some ${\alpha}{\in}{\Gamma}$ if $e{\alpha}e$ = e. A ${\Gamma}$-semigroup S is called an E-inversive ${\Gamma}$-semigroup if for each $a{\in}S$ there exist $x{\in}S$ and ${\alpha}{\in}{\Gamma}$ such that a${\alpha}$x is a ${\beta}$-idempotent for some ${\beta}{\in}{\Gamma}$. A ${\Gamma}$-semigroup is called a right E-${\Gamma}$-semigroup if for each ${\alpha}$-idempotent e and ${\beta}$-idempotent f, $e{\alpha}$ is a ${\beta}$-idempotent. In this paper we investigate different properties of E-inversive ${\Gamma}$-semigroup and right E-${\Gamma}$-semigroup.

ON CLOSURE GAMMA-SEMIGROUPS

  • Jun, Young-Bae
    • Communications of the Korean Mathematical Society
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    • v.19 no.4
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    • pp.639-641
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    • 2004
  • We introduce the notion of closure $\Gamma$-semigroups. We give a condition for a closure $\Gamma$-semigroup to be $\Gamma$-central, and we show that the $\Gamma$-centralizer of a closure $\Gamma$-semigroup is a $\Gamma$-subsemigroup.

THE CLASS GROUP OF D*/U FOR D AN INTEGRAL DOMAIN AND U A GROUP OF UNITS OF D

  • Chang, Gyu Whan
    • Korean Journal of Mathematics
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    • v.17 no.2
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    • pp.189-196
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    • 2009
  • Let D be an integral domain, and let U be a group of units of D. Let $D^*=D-\{0\}$ and ${\Gamma}=D^*/U$ be the commutative cancellative semigroup under aU+bU=abU. We prove that $Cl(D)=Cl({\Gamma})$ and that D is a PvMD (resp., GCD-domain, Mori domain, Krull domain, factorial domain) if and only if ${\Gamma}$ is a PvMS(resp., GCD-semigroup, Mori semigroup, Krull semigroup, factorial semigroup). Let U=U(D) be the group of units of D. We also show that if D is integrally closed, then $D[{\Gamma}]$, the semigroup ring of ${\Gamma}$ over D, is an integrally closed domain with $Cl(D[{\Gamma}])=Cl(D){\oplus}Cl(D)$; hence D is a PvMD (resp., GCD-domain, Krull domain, factorial domain) if and only if $D[{\Gamma}]$ is.

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GENERALIZED IDEAL ELEMENTS IN le-Γ-SEMIGROUPS

  • Hila, Kostaq;Pisha, Edmond
    • Communications of the Korean Mathematical Society
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    • v.26 no.3
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    • pp.373-384
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    • 2011
  • In this paper we introduce and give some characterizations of (m, n)-regular le-${\Gamma}$-semigroup in terms of (m, n)-ideal elements and (m, n)-quasi-ideal elements. Also, we give some characterizations of subidempotent (m, n)-ideal elements in terms of $r_{\alpha}$- and $l_{\alpha}$- closed elements.

ON WEAKLY PRIME IDEALS OF ORDERED ${\gamma}$-SEMIGROUPS

  • Kwon, Young-In;Lee, Sang-Keun
    • 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.

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ON REES MATRIX REPRESENTATIONS OF ABUNDANT SEMIGROUPS WITH ADEQUATE TRANSVERSALS

  • Gao, Zhen Lin;Liu, Xian Ge;Xiang, Yan Jun;Zuo, He Li
    • Communications of the Korean Mathematical Society
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    • v.24 no.4
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    • pp.481-500
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    • 2009
  • The concepts of *-relation of a ($\Gamma$-)semigroup and $\bar{\Gamma}$-adequate transversal of a ($\Gamma$-)abundant semigroup are defined in this note. Then we develop a matrix type theory for abundant semigroups. We give some equivalent conditions of a Rees matrix semigroup being abundant and some equivalent conditions of an abundant Rees matrix semigroup having an adequate transversal. Then we obtain some Rees matrix representations for abundant semigroups with adequate transversals by the above theories.

COINCIDENCES OF DIFFERENT TYPES OF FUZZY IDEALS IN ORDERED Γ-SEMIGROUPS

  • Kanlaya, Arunothai;Iampan, Aiyared
    • Korean Journal of Mathematics
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    • v.22 no.2
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    • pp.367-381
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    • 2014
  • The notion of ${\Gamma}$-semigroups was introduced by Sen in 1981 and that of fuzzy sets by Zadeh in 1965. Any semigroup can be reduced to a ${\Gamma}$-semigroup but a ${\Gamma}$-semigroup does not necessarily reduce to a semigroup. In this paper, we study the coincidences of fuzzy generalized bi-ideals, fuzzy bi-ideals, fuzzy interior ideals and fuzzy ideals in regular, left regular, right regular, intra-regular, semisimple ordered ${\Gamma}$-semigroups.

ON THE ORDERED n-PRIME IDEALS IN ORDERED Γ-SEMIGROUPS

  • Siripitukdet, Manoj;Iampan, Aiyared
    • 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.