• 대한수학회논문집
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• 제37권3호
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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.

• 대한수학회논문집
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• 제35권4호
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• pp.1057-1073
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• 2020

In this paper, we study some of the factorization aspects of rational multicyclic monoids, that is, additive submonoids of the nonnegative rational numbers generated by multiple geometric sequences. In particular, we provide a complete description of the rational multicyclic monoids M that are hereditarily atomic (i.e., every submonoid of M is atomic). Additionally, we show that the sets of lengths of certain rational multicyclic monoids are finite unions of multidimensional arithmetic progressions, while their unions satisfy the Structure Theorem for Unions of Sets of Lengths. Finally, we realize arithmetic progressions as the sets of distances of some additive submonoids of the nonnegative rational numbers.

• Korean Journal of Mathematics
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• 제27권2호
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• pp.535-545
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• 2019

The main idea of this paper is to introduce the notion of $cat^1$-monoids and to prove that the category of crossed semimodules ${\mathcal{C}}=(A,B,{\partial})$ where A is a group is equivalent to the category of $cat^1$-monoids. This is a generalization of the well known equivalence between category of $cat^1$-groups and that of crossed modules over groups.

• 대한수학회논문집
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• 제39권1호
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• pp.259-266
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• 2024

The purpose of this note is a wide generalization of the topological results of various classes of ideals of rings, semirings, and modules, endowed with Zariski topologies, to r-strongly irreducible r-ideals (endowed with Zariski topologies) of monoids, called terminal spaces. We show that terminal spaces are T₀, quasi-compact, and every nonempty irreducible closed subset has a unique generic point. We characterize rarithmetic monoids in terms of terminal spaces. Finally, we provide necessary and sufficient conditions for the subspaces of r-maximal r-ideals and r-prime r-ideals to be dense in the corresponding terminal spaces.

• 대한수학회지
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• 제52권1호
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• pp.1-21
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• 2015

C-domains are defined via class semigroups, and every C-domain is a Mori domain with nonzero conductor whose complete integral closure is a Krull domain with finite class group. In order to extend the concept of C-domains to rings with zero divisors, we study v-Marot rings as generalizations of ordinary Marot rings and investigate their theory of regular divisorial ideals. Based on this we establish a generalization of a result well-known for integral domains. Let R be a v-Marot Mori ring, $\hat{R}$ its complete integral closure, and suppose that the conductor f = (R : $\hat{R}$) is regular. If the residue class ring R/f and the class group C($\hat{R}$) are both finite, then R is a C-ring. Moreover, we study both v-Marot rings and C-rings under various ring extensions.

• 대한수학회논문집
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• 제39권2호
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• pp.313-329
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• 2024

An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of atoms (counting repetitions) in the corresponding sum is called the length of the factorization. Following Geroldinger and Zhong, we say that an atomic monoid M is a length-finite factorization monoid if each b ∈ M has only finitely many factorizations of any prescribed length. An additive submonoid of ℝ_≥0 is called a positive monoid. Factorizations in positive monoids have been actively studied in recent years. The main purpose of this paper is to give a better understanding of the non-unique factorization phenomenon in positive monoids through the lens of the length-finite factorization property. To do so, we identify a large class of positive monoids which satisfy the length-finite factorization property. Then we compare the length-finite factorization property to the bounded and the finite factorization properties, which are two properties that have been systematically investigated for more than thirty years.

• 대한수학회보
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• 제53권2호
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• pp.495-506
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• 2016

In this note we consider the monoid $\mathcal{PODI}_n$ of all monotone partial permutations on $\{1,{\ldots},n\}$ and its submonoids $\mathcal{DP}_n$, $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ of all partial isometries, of all order-preserving partial permutations and of all order-preserving partial isometries, respectively. We prove that both the monoids $\mathcal{POI}_n$ and $\mathcal{ODP}_n$ are quotients of bilateral semidirect products of two of their remarkable submonoids, namely of extensive and of co-extensive transformations. Moreover, we show that $\mathcal{PODI}_n$ is a quotient of a semidirect product of $\mathcal{POI}_n$ and the group $\mathcal{C}_2$ of order two and, analogously, $\mathcal{DP}_n$ is a quotient of a semidirect product of $\mathcal{ODP}_n$ and $\mathcal{C}_2$.

• 대한수학회논문집
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• 제32권2호
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• pp.233-260
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• 2017

Let R be a factorial domain. In this work we investigate the connections between the arithmetic of Int(R) (i.e., the ring of integer-valued polynomials over R) and its monadic submonoids (i.e., monoids of the form {$g{\in}Int(R){\mid}g{\mid}_{Int(R)}f^k$ for some $k{\in}{\mathbb{N}}_0$} for some nonzero $f{\in}Int(R)$). Since every monadic submonoid of Int(R) is a Krull monoid it is possible to describe the arithmetic of these monoids in terms of their divisor-class group. We give an explicit description of these divisor-class groups in several situations and provide a few techniques that can be used to determine them. As an application we show that there are strong connections between Int(R) and its monadic submonoids. If $R={\mathbb{Z}}$ or more generally if R has sufficiently many "nice" atoms, then we prove that the infinitude of the elasticity and the tame degree of Int(R) can be explained by using the structure of monadic submonoids of Int(R).

• 대한수학회논문집
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• 제35권3호
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• pp.697-709
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• 2020

In this paper, the class of quasi-annihilator (homo)flat acts based on the notion of quasi-annihilator ideal is introduced. This class lies strictly between the classes of weakly (homo)flat and principally weakly (homo)flat acts. Some properties of such kinds of flatness are studied. We present some homological classifications for monoids by means of quasiannihilator (homo)flatness of their right Rees factor acts.

• 대한수학회논문집
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• 제37권2호
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• pp.347-358
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• 2022

In this paper, we first introduce the notions of superfluous and coessential subacts. Then hollow and co-uniform S-acts are defined as the acts that all proper subacts are superfluous and coessential, respectively. Also it is indicated that the class of hollow S-acts is properly between two classes of indecomposable and locally cyclic S-acts. Moreover, using the notion of radical of an S-act as the intersection of all maximal subacts, the relations between hollow and local S-acts are investigated. Ultimately, the notion of a supplement of a subact is defined to characterize the union of hollow S-acts.