• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1207-1218
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• 2011

Let * be a star-operation on an integral domain R, and let $\mathfrak{I}_*^+(R)$ be the semigroup of *-invertible integral *-ideals of R. In this article, we introduce the concept of a *-coatom, and we then characterize when $\mathfrak{I}_*^+(R)$ is a free semigroup with a set of free generators consisting of *-coatoms. In particular, we show that $\mathfrak{I}_*^+(R)$ is a free semigroup if and only if R is a Krull domain and each ${\upsilon}$-invertible ${\upsilon}$-ideal is *-invertible. As a corollary, we obtain some characterizations of *-Dedekind domains.

• Honam Mathematical Journal
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• v.46 no.3
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• pp.500-514
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• 2024

This paper explores the concepts of quasi covered ideal, quasi base and the greatest quasi covered ideal within the context of an ordered semigroup, extending the study of algebraic structures to incorporate both the algebraic and order theoretic perspectives. An ordered semigroup provides a rich framework for investigating the interplay between algebraic and order structure. Also, we provide the conditions for the greatest ideal to be quasi covered ideal and develop the fundamental properties with implications of quasi covered ideal of an ordered semigroup. Moreover, we study the relationship between covered ideal with quasi covered ideal, greatest ideal with quasi covered ideal and the greatest quasi covered ideal with quasi base of an ordered semigroup.

• Honam Mathematical Journal
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• v.8 no.1
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• pp.1-19
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• 1986

We show that a (possibly unbounded) linear operator, T, is scalar on the real line (spectral operator of scalar type, with real spectrum) if and only if (iT) generates a uniformly bounded semigroup and $(1-iT)(1+iT)^{-1}$ is scalar on the unit circle. T is scalar on [0, $\infty$) if and only if T generates a uniformly bounded semigroup and $(1+T)^{-1}$ is scalar on [0,1). By analogy with these results, we define $C^0$-scalar, on the real line, or [0. $\infty$), for an unbounded operator. We show that a generator of a positive-definite group is $C^0$-scalar on the real line. and a generator of a completely monotone semigroup is $C^0$-scalar on [0, $\infty$). We give sufficient conditions for a closed operator, T, to generate a positive-definite group: the sequence < $\phi(T^{n}x)$ > $_{n=0}^{\infty}$ must equal the moments of a positive measure on the real line, for sufficiently many positive $\phi$ in $X^{*}$, x in X. If the measures are supported on [0, $\infty$), then T generates a completely monotone semigroup. On a reflexive Banach lattice, these conditions are also necessary, and are equivalent to T being scalar, with positive projection-valued measure. T generates a completely monotone semigroup if and only if T is positive and m-dispersive and generates a bounded holomorphic semigroup.

• Bulletin of the Korean Mathematical Society
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• v.49 no.1
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• pp.109-126
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• 2012

In 2003, Marques-Smith and Sullivan described the join ${\Omega}$ of the 'natural order' $\leq$ and the 'containment order' $\subseteq$ on P(X), the semigroup under composition of all partial transformations of a set X. And, in 2004, Pinto and Sullivan described all automorphisms of PS(q), the partial Baer-Levi semigroup consisting of all injective ${\alpha}{\in}P(X)$ such that ${\mid}X{\backslash}X{\alpha}\mid=q$, where $N_0{\leq}q{\leq}{\mid}X{\mid}$. In this paper, we describe the group of automorphisms of R(q), the largest regular subsemigroup of PS(q). In 2010, we studied some properties of $\leq$ and $\subseteq$ on PS(q). Here, we characterize the meet and join under those orders for elements of R(q) and PS(q). In addition, since $\leq$ does not equal ${\Omega}$ on I(X), the symmetric inverse semigroup on X, we formulate an algebraic version of ${\Omega}$ on arbitrary inverse semigroups and discuss some of its properties in an algebraic setting.

• Kyungpook Mathematical Journal
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• v.34 no.1
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• pp.53-57
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• 1994

• Journal of applied mathematics & informatics
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• v.30 no.3_4
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• pp.677-683
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• 2012

W.Choi([1]) obtains a complete description of ergodic property and several property by making use of the semigroup method. In this note, we shall consider separately the martingale problems for two operators A and B as a detail decomposition of operator L. A key point is that the (K, L, $p$)-martingale problem in population genetics model is related to diffusion processes, so we begin with some a priori estimates and we shall show existence of contraction semigroup {$T_t$} associated with decomposition operator A.

• Communications of the Korean Mathematical Society
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• v.27 no.4
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• pp.689-699
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• 2012

(S, *) is a semigroup S equipped with a unary operation "*". This work is devoted to a class of unary semigroups, namely E-$inversive$ *-$semigroups$. A unary semigroup (S, *) is called an E-inversive *-semigroup if the following identities hold: $$x^*xx^*=x^*$$, $$(x^*)^*=xx^*x$$, $$(xy)^*=y^*x^*$$. In this paper, E-inversive *-semigroups are characterized by several methods. Furthermore, congruences on these semigroups are also studied.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.1
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• pp.27-37
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• 2000

In this paper, we will introduce the pan-dual generalized fuzzy integral of a commutative isotonic semigroup-valued functions, which is generalized of the (DG) fuzzy integral and investigate the fundamental properties of this kind of fuzzy integral.

• Bulletin of the Korean Mathematical Society
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• v.50 no.2
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• pp.627-637
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• 2013

Let [$n$] = {1, 2, ${\ldots}$, $n$} be ordered in the standard way. The order-preserving full transformation semigroup ${\mathcal{O}}_n$ is the set of all order-preserving singular full transformations on [$n$] under composition. For this semigroup we describe maximal subsemibands, maximal regular subsemibands, locally maximal regular subsemibands, and completely obtain their classification.

• Bulletin of the Korean Mathematical Society
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• v.35 no.2
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• pp.259-268
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• 1998

We give several characterizations of a TV-PVMD and we show that the localization R[X;S]$_{N_{\upsilon}}$ of a semigroup ring R[X;S] is a TV-PVMD if and only if R is a TV-PVMD where $N_{\upsilon}\;=\;\{f\;{\in}\;R[X]{\mid}(A_f)_{\upsilon} = R\}$ and S is a torsion free cancellative semigroup with zero.