• 한국수학교육학회지시리즈B:순수및응용수학
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• 제30권4호
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• pp.377-395
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• 2023

The purpose of this paper is to consider the abstract theory of hypernear rings. In this regard, we derive the isomorphism theorems for hypernear rings as well as Chinese Remainder theorem. Our results can be considered as a generalization for the cases of Krasner hyperrings, near rings and rings.

• Kyungpook Mathematical Journal
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• 제13권2호
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• pp.235-240
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• 1973

A k-dimensional vector bundle is a bundle ${\xi}=(E,P,B,F^k)$ with fibre $F^k$ satisfying the local triviality, where F is the field of real numbers R or complex numbers C ([1], [2] and [3]). Let $Vect_k(X)$ be the set consisting of all isomorphism classes of k-dimensional vector bundles over the topological space X. Then $Vect_F(X)=\{Vect_k(X)\}_{k=0,1,{\cdots}}$ is a semigroup with Whitney sum (${\S}1$). For a pair (X, A) of topological spaces, a difference isomorphism over (X, A) is a vector bundle morphism ([2], [3]) ${\alpha}:{\xi}_0{\rightarrow}{\xi}_1$ such that the restriction ${\alpha}:{\xi}_0{\mid}A{\longrightarrow}{\xi}_1{\mid}A$ is an isomorphism. Let $S_k(X,A)$ be the set of all difference isomorphism classes over (X, A) of k-dimensional vector bundles over X with fibre $F^k$. Then $S_F(X,A)=\{S_k(X,A)\}_{k=0,1,{\cdots}}$, is a semigroup with Whitney Sum (${\S}2$). In this paper, we shall prove a relation between $Vect_F(X)$ and $S_F(X,A)$ under some conditions (Theorem 2, which is the main theorem of this paper). We shall use the following theorem in the paper. THEOREM 1. Let ${\xi}=(E,P,B)$ be a locally trivial bundle with fibre F, where (B, A) is a relative CW-complex. Then all cross sections S of ${\xi}{\mid}A$ prolong to a cross section $S^*$ of ${\xi}$ under either of the following hypothesis: (H1) The space F is (m-1)-connected for each $m{\leq}dim$ B. (H2) There is a relative CW-complex (Y, X) such that $B=Y{\times}I$ and $A=(X{\times}I)$ ${\cap}(Y{\times}O)$, where I=[0, 1]. (For proof see p.21 [2]).

• Kyungpook Mathematical Journal
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• 제61권2호
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• pp.223-237
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• 2021

Let n be a fixed natural number. Menger algebras of rank n can be regarded as a canonical generalization of arbitrary semigroups. This paper is concerned with studying algebraic properties of Menger algebras of rank n by first defining a special class of full n-place functions, the so-called a left translation, which possess necessary and sufficient conditions for an (n + 1)-groupoid to be a Menger algebra of rank n. The isomorphism parts begin with introducing the concept of homomorphisms, and congruences in Menger algebras of rank n. These lead us to establish a quotient structure consisting a nonempty set factored by such congruences together with an operation defined on its equivalence classes. Finally, the fundamental homomorphism theorem and isomorphism theorems for Menger algebras of rank n are given. As a consequence, our results are significant in the study of algebraic theoretical Menger algebras of rank n. Furthermore, we extend the usual notions of ordinary semigroups in a natural way.

• Korean Journal of Mathematics
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• 제26권3호
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• pp.373-385
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• 2018

In this paper we introduce the notion of a fuzzy kernel of a fuzzy homomorphism on groups and we show that it is a fuzzy normal subgroup of the domain group. Conversely, we also prove that any fuzzy normal subgroup is a fuzzy kernel of some fuzzy epimorphism, namely the canonical fuzzy epimorphism. Finally, we formulate and prove the fuzzy version of the fundamental theorem of homomorphism and those isomorphism theorems.

• 전기전자학회논문지
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• 제3권2호
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• pp.295-304
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• 1999

본 논문에서는 유한체상에서의 다치논리시스템구성을 그래프이론에 기초를 둔 결정도에 의해 구성하는 방법을 제안하였다. 제안한 방법은 먼저 다치논리 Shannon의 확장전개를 토대로 다치논리결정도를 도출하였으며, 부그래프를 적용하여 함수분할을 수행하였다. 그리고 각종 그래프의 동형관계와 정점의 재순서화를 적용하여 결정도의 변수순서선텍알고리즘과 간략화 알고리즘을 제안하였으며 이로부터 최종 다치논리시스템을 설계하는 방법을 제안하였다.

• 한국수학교육학회지시리즈A:수학교육
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• 제14권1호
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• pp.22-26
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• 1975

In this paper, we define a differential module and study its properties. In section 2, as for propositions, Ive research some properties, directsum, isomorphism of factorization, exact sequence of derived modules. And then as for theorem, I try to present the following statement, if the sequence of homomorphisms of differential modules is exact. Then the sequence of homomorphisms of Z(X) is exact, also the sequence of homomorphisms of Z(X) is exact. According to the theorem, as for Lemma, we consider commutative diagram between exact sequence of Z(X) and exact sequence of Z'(X) . As an immediate consequence of this theorem, we obtain the following result. If M is an arbitrary module and the sequence of homomorphisms of the modules Z(X) is exact, then the sequence of their tensor products with the trivial endomorphism is semi-exact.

• 대한수학회논문집
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• 제33권3호
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• pp.705-710
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• 2018

Let T(X) be the full transformation semigroup on a set X and ${\sigma}$ be an equivalence relation on X. Denote $$E(X,{\sigma})=\{{\alpha}{\in}T(X):{\forall}x,\;y{\in}X,\;(x,y){\in}{\sigma}\;\text{implies}\;x{\alpha}=y{\alpha}\}.$$. Then $E(X,{\sigma})$ is a subsemigroup of T(X). In this paper, we characterize two semigroups of type $E(X,{\sigma})$ when they are isomorphic.

• 대한수학회지
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• 제45권2호
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• pp.527-535
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• 2008

Sufficient conditions are given to assert that between any two Banach spaces over $\mathbb{K}$ Fredholm mappings share exactly N values in a specific open ball. The proof of the result is constructive and is based upon continuation methods.

• 한국지능시스템학회논문지
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• 제21권4호
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• pp.506-511
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• 2011

In this paper we investigate some situations in connection with two exact sequences of fuzzy linear maps. Also we obtain a generalization of the work [Theorem4] of Pan [5], and we study the analogies of The Four Lemma and The Five Lemma of homological algebra. Finally we obtain a special exact sequence.

• 대한수학회보
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• 제32권1호
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• pp.85-91
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• 1995

For a locally compact Hausdorff space, we denote by $C_0(X)$ the Banach space of all continuous complex valued functions defined on X which vanish at infinity, equipped with the usual sup norm. In case X is compact, we write C(X) instead of $C_0(X)$. A well-known Banach-Stone theorem states that the existence of an isometry between the function spaces $C_0(X)$ and $C_0(Y)$ implies X and Y are homemorphic. D. Amir [1] and M. Cambern [2] independently generalized this theorem by proving that if $C_0(X)$ and $C_0(Y)$ are isomorphic under an isomorphism T satisfying $\left\$\mid$ T \right\$\mid$ \left\$\mid$ T^1 \right\$\mid$ < 2$, then X and Y must also be homeomorphic.