• Title/Summary/Keyword: I-semiring

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On Partitioning and Subtractive Ideals of Ternary Semirings

  • Chaudhari, Jaiprakash Ninu;Ingale, Kunal Julal
    • Kyungpook Mathematical Journal
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    • v.51 no.1
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    • pp.69-76
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    • 2011
  • In this paper, we introduce a partitioning ideal of a ternary semiring which is useful to develop the quotient structure of ternary semiring. Indeed we prove : 1) The quotient ternary semiring S/$I_{(Q)}$ is essentially independent of choice of Q. 2) If f : S ${\rightarrow}$ S' is a maximal ternary semiring homomorphism, then S/ker $f_{(Q)}$ ${\cong}$ S'. 3) Every partitioning ideal is subtractive. 4) Let I be a Q-ideal of a ternary semiring S. Then A is a subtractive ideal of S with I ${\subseteq}$ A if and only if A/$I_{(Q{\cap}A)}$ = {q + I : q ${\in}$ Q ${\cap}$ A} is a subtractive idea of S/$I_{(Q)}$.

A NOTE ON DERIVATIONS OF ORDERED 𝚪-SEMIRINGS

  • Kim, Kyung Ho
    • Korean Journal of Mathematics
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    • v.27 no.3
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    • pp.779-791
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    • 2019
  • In this paper, we consider derivation of an ordered ${\Gamma}$-semiring and introduce the notion of reverse derivation on ordered ${\Gamma}$-semiring. Also, we obtain some interesting related properties. Let I be a nonzero ideal of prime ordered ${\Gamma}$-semiring M and let d be a nonzero derivation of M. If ${\Gamma}$-semiring M is negatively ordered, then d is nonzero on I.

TOTAL IDENTITY-SUMMAND GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO A CO-IDEAL

  • Atani, Shahabaddin Ebrahimi;Hesari, Saboura Dolati Pish;Khoramdel, Mehdi
    • Journal of the Korean Mathematical Society
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    • v.52 no.1
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    • pp.159-176
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    • 2015
  • Let R be a semiring, I a strong co-ideal of R and S(I) the set of all elements of R which are not prime to I. In this paper we investigate some interesting properties of S(I) and introduce the total identity-summand graph of a semiring R with respect to a co-ideal I. It is the graph with all elements of R as vertices and for distinct x, $y{\in}R$, the vertices x and y are adjacent if and only if $xy{\in}S(I)$.

TOTAL GRAPH OF A COMMUTATIVE SEMIRING WITH RESPECT TO IDENTITY-SUMMAND ELEMENTS

  • Atani, Shahabaddin Ebrahimi;Hesari, Saboura Dolati Pish;Khoramdel, Mehdi
    • Journal of the Korean Mathematical Society
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    • v.51 no.3
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    • pp.593-607
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    • 2014
  • Let R be an I-semiring and S(R) be the set of all identity-summand elements of R. In this paper we introduce the total graph of R with respect to identity-summand elements, denoted by T(${\Gamma}(R)$), and investigate basic properties of S(R) which help us to gain interesting results about T(${\Gamma}(R)$) and its subgraphs.

On 2-Absorbing and Weakly 2-Absorbing Ideals of Commutative Semirings

  • Darani, Ahmad Yousefian
    • Kyungpook Mathematical Journal
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    • v.52 no.1
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    • pp.91-97
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    • 2012
  • Let $R$ be a commutative semiring. We define a proper ideal $I$ of $R$ to be 2-absorbing (resp., weakly 2-absorbing) if $abc{\in}I$ (resp., $0{\neq}abc{\in}I$) implies $ab{\in}I$ or $ac{\in}I$ or $bc{\in}I$. We show that a weakly 2-absorbing ideal $I$ with $I^3{\neq}0$ is 2-absorbing. We give a number of results concerning 2-absorbing and weakly 2-absorbing ideals and examples of weakly 2-absorbing ideals. Finally we de ne the concept of 0 - (1-, 2-, 3-)2-absorbing ideals of $R$ and study the relationship among these classes of ideals of $R$.

A CHARACTERIZATION OF FINITE FACTORIZATION POSITIVE MONOIDS

  • Polo, Harold
    • Communications of the Korean Mathematical Society
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    • v.37 no.3
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    • pp.669-679
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    • 2022
  • We provide a characterization of the positive monoids (i.e., additive submonoids of the nonnegative real numbers) that satisfy the finite factorization property. As a result, we establish that positive monoids with well-ordered generating sets satisfy the finite factorization property, while positive monoids with co-well-ordered generating sets satisfy this property if and only if they satisfy the bounded factorization property.

On 2-Absorbing and Weakly 2-Absorbing Primary Ideals of a Commutative Semiring

  • Soheilnia, Fatemeh
    • Kyungpook Mathematical Journal
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    • v.56 no.1
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    • pp.107-120
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    • 2016
  • Let R be a commutative semiring. The purpose of this note is to investigate the concept of 2-absorbing (resp., weakly 2-absorbing) primary ideals generalizing of 2-absorbing (resp., weakly 2-absorbing) ideals of semirings. A proper ideal I of R said to be a 2-absorbing (resp., weakly 2-absorbing) primary ideal if whenever $a,b,c{\in}R$ such that $abc{\in}I$ (resp., $0{\neq}abc{\in}I$), then either $ab{\in}I$ or $bc{\in}\sqrt{I}$ or $ac{\in}\sqrt{I}$. Moreover, when I is a Q-ideal and P is a k-ideal of R/I with $I{\subseteq}P$, it is shown that if P is a 2-absorbing (resp., weakly 2-absorbing) primary ideal of R, then P/I is a 2-absorbing (resp., weakly 2-absorbing) primary ideal of R/I and it is also proved that if I and P/I are weakly 2-absorbing primary ideals, then P is a weakly 2-absorbing primary ideal of R.

THE IDENTITY-SUMMAND GRAPH OF COMMUTATIVE SEMIRINGS

  • Atani, Shahabaddin Ebrahimi;Hesari, Saboura Dolati Pish;Khoramdel, Mehdi
    • Journal of the Korean Mathematical Society
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    • v.51 no.1
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    • pp.189-202
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    • 2014
  • An element r of a commutative semiring R with identity is said to be identity-summand if there exists $1{\neq}a{\in}R$ such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by ${\Gamma}(R)$. It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph ${\Gamma}(R)$ are studied.

ON DIFFERENTIAL IDENTITIES INVOLVING PARTITIONING IDEALS OF SEMIRINGS

  • Liaqat Ali;Muhammad Aslam;Ghulam Farid;Tariq Mahmood
    • Communications of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.595-609
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    • 2024
  • In this article, we study a certain class of partitioning ideals known as Q-ideals, in semirings. Main objective is to investigate differential identities linking a semiring S to its prime Q-ideal IQ, which ensure the commutativity and other features of S/IQ.

STRUCTURES OF IDEMPOTENT MATRICES OVER CHAIN SEMIRINGS

  • Kang, Kyung-Tae;Song, Seok-Zun;Yang, Young-Oh
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.721-729
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    • 2007
  • In this paper, we have characterizations of idempotent matrices over general Boolean algebras and chain semirings. As a consequence, we obtain that a fuzzy matrix $A=[a_{i,j}]$ is idempotent if and only if all $a_{i,j}$-patterns of A are idempotent matrices over the binary Boolean algebra $\mathbb{B}_1={0,1}$. Furthermore, it turns out that a binary Boolean matrix is idempotent if and only if it can be represented as a sum of line parts and rectangle parts of the matrix.