• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.617-629
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• 2019

A term is called linear if each variable which occurs in the term, occurs only once. A hypersubstitution is said to be linear if it maps any operation symbol to a linear term of the same arity. Linear hypersubstitutions have some importance in Theoretical Computer Science since they preserve recognizability [7]. We show that the collection of all linear hypersubstitutions forms a monoid. Linear hypersubstitutions are used to define linear hyperidentities. The set of all linear term operations of a given algebra forms with respect to certain superposition operations a function algebra. Hypersubstitutions define endomorphisms on this function algebra.

• Korean Journal of Mathematics
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• v.19 no.3
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• pp.255-261
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• 2011

Let D be an integral domain, S be a torsion-free grading monoid such that the quotient group of S is of type (0, 0, 0, ${\ldots}$), and D[S] be the semigroup ring of S over D. We show that D[S] is an H-domain if and only if D is an H-domain and each maximal t-ideal of S is a $v$-ideal. We also show that if $\mathbb{R}$ is the eld of real numbers and if ${\Gamma}$ is the additive group of rational numbers, then $\mathbb{R}[{\Gamma}]$ is not an H-domain.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.337-342
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• 1995

A semiring is a binary system $(S, +, \times)$ such that (S, +) is an Abelian monoid (identity 0), (S,x) is a monoid (identity 1), $\times$ distributes over +, 0 $\times s s \times 0 = 0$ for all s in S, and $1 \neq 0$. Usually S denotes the system and $\times$ is denoted by juxtaposition. If $(S,\times)$ is Abelian, then S is commutative. Thus all rings are semirings. Some examples of semirings which occur in combinatorics are Boolean algebra of subsets of a finite set (with addition being union and multiplication being intersection) and the nonnegative integers (with usual arithmetic). The concepts of matrix theory are defined over a semiring as over a field. Recently a number of authors have studied various problems of semiring matrix theory. In particular, Minc [4] has written an encyclopedic work on nonnegative matrices.

• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.139-143
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• 2011

In this paper we consider mappings ${\sigma}$ which map the binary operation symbol f to the term ${\sigma}$(f) which do not necessarily preserve the arities. We call these mappings generalized hypersubstitutions. Any generalized hypersubstitution ${\sigma}$ can be extended to a mapping $\hat{\sigma}$ on the set of all terms of type ${\tau}$ = (2). We de ne a binary operation on the set $Hyp_G$(2) of all generalized hypersubstitutions of type ${\tau}$ = (2) by using this extension The set $Hyp_G$(2) together with the identity generalized hypersubstitution ${\sigma}_{id}$ which maps f to the term f($x_1,x_2$) forms a monoid. We determine all regular elements of this monoid.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.363-377
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• 2015

Let R be a ring, (S,${\leq}$) a strictly ordered monoid and ${\omega}$ : S ${\rightarrow}$ End(R) a monoid homomorphism. The skew generalized power series ring R[[S,${\omega}$]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we investigate the interplay between the ring-theoretical properties of R[[S,${\omega}$]] and the graph-theoretical properties of its zero-divisor graph ${\Gamma}$(R[[S,${\omega}$]]). Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.

• Korean Journal of Mathematics
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• v.31 no.4
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• pp.445-463
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• 2023

The paper introduces the concept of direct product and discusses some basic facts about distant ideals. We also introduce the definition of directly indecomposable in an autometrized algebra. Furthermore, we present the concept of a subdirect product and simple autometrized algebra and its behavior. We also introduce the definition of subdirectly irreducible in an autometrized algebras. In particular, we prove that every subdirectly irreducible monoid autometrized algebra is directly indecomposable. Finally, we discuss different properties of chain autometrized algebras and introduce the representability in the autometrized algebra. We also prove that if a weak chain monoid normal autometrized l-algebra is nilradical, then it is representable.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.489-498
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• 2009

We obtain two new characterizations of separativity of refinement monoids, in terms of comparability-type conditions. As applications, we get several equivalent conditions of separativity for exchange ideals.

• Korean Journal of Mathematics
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• v.10 no.2
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• pp.149-155
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• 2002

We develop some basic properties of $t$-fuzzy ideals in a monoid or a group and find the sufficient conditions for a fuzzy set in a division ring to be a $t$-fuzzy subring and the necessary and sufficient conditions for a fuzzy set in a division ring to be a $t$-fuzzy ideal.

• East Asian mathematical journal
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• v.22 no.2
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• pp.189-194
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• 2006

Throughout this paper, a semigroup S will denote a torsion free grading monoid, and it is a non-zero semigroup with 0. The operation is written additively. The aim of this paper is to study semigroup version of an integral domain ([1],[3],[4] and [5]).

• East Asian mathematical journal
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• v.27 no.3
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• pp.309-317
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• 2011

In this paper we introduce an algorithm to construct an in-finite family of magic cubes and investigate properties of magic cubes, which extend the results in [6].