• Korean Journal of Mathematics
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• 제27권2호
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• pp.547-562
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• 2019

In this paper, we define a local topological group-groupoid and prove that if G is a local topological group-groupoid, then the monodromy groupoid Mon(G) of G is a local group-groupoid.

• 대한수학회보
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• 제21권2호
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• pp.67-75
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• 1984

Let X be a topological space. We consider a groupoid G over X and the quotient groupoid G/N for any normal subgroupoid N of G. The concept of groupoid (topological groupoid) is a natural generalization of the group(topological group). An useful example of a groupoid over X is the foundamental groupoid .pi.X whose object group at x.mem.X is the fundamental group .pi.(X, x). It is known [5] that if X is locally simply connected, then the topology of X determines a topology on .pi.X so that is becomes a topological groupoid over X, and a covering space of the product space X*X. In this paper the concept of the locally simple connectivity of a topological space X is applied to the groupoid G over X. That concept is defined as a term '1-connected local subgroupoid' of G. Using this concept we topologize the groupoid G so that it becomes a topological groupoid over X. With this topology the connected groupoid G is a covering space of the product space X*X. Further-more, if ob(.overbar.G)=.overbar.X is a covering space of X, then the groupoid .overbar.G is also a covering space of the groupoid G. Since the fundamental groupoid .pi.X of X satisfying a certain condition has an 1-connected local subgroupoid, .pi.X can always be topologized. In this case the topology on .pi.X is the same as that of [5]. In section 4 the results on the groupoid G are generalized to the quotient groupoid G/N. For any topological groupoid G over X and normal subgroupoid N of G, the abstract quotient groupoid G/N can be given the identification topology, but with this topology G/N need not be a topological groupoid over X [4]. However the induced topology (H) on G makes G/N (with the identification topology) a topological groupoid over X. A final section is related to the covering morphism. Let G$_{1}$ and G$_{2}$ be groupoids over the sets X$_{1}$ and X$_{2}$, respectively, and .phi.:G$_{1}$.rarw.G$_{2}$ be a covering spimorphism. If X$_{2}$ is a topological space and G$_{2}$ has an 1-connected local subgroupoid, then we can topologize X$_{1}$ so that ob(.phi.):X$_{1}$.rarw.X$_{2}$ is a covering map and .phi.: G$_{1}$.rarw.G$_{2}$ is a topological covering morphism.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제19권2호
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• pp.199-209
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• 2012

For a locally compact higher rank graph ${\Lambda}$, we construct a two-sided path space ${\Lambda}^{\Delta}$ with shift homeomorphism ${\sigma}$ and its corresponding path groupoid ${\Gamma}$. Then we find equivalent conditions of aperiodicity, cofinality and irreducibility of ${\Lambda}$ in (${\Lambda}^{\Delta}$, ${\sigma}$), ${\Gamma}$, and the groupoid algebra $C^*({\Gamma})$.

• Korean Journal of Mathematics
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• 제23권3호
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• pp.357-370
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• 2015

In this paper, we introduce the concept of (m, n)-ideals in a non-associative ordered structure, which is called an ordered Abel-Grassmann's groupoid, by generalizing the concept of (m, n)-ideals in an ordered semigroup [14]. We also study the (m, n)-regular class of an ordered AG-groupoid in terms of (m, n)-ideals.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제18권3호
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• pp.261-268
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• 2011

For a compact Hausdorff space X with its p-fold covering map ${\sigma}$, we construct its corresponding topological groupoid ${\Gamma}$, and show that there is a strong relation between the dynamical structures of (X, ${\sigma}$) and the groupoid structures of ${\Gamma}$.

• Korean Journal of Mathematics
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• 제13권1호
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• pp.13-18
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• 2005

Recently, Kehayopulu and Tsingelis studied for prime ideals of groupoids-ordered groupoids. In this paper, we give some results on prime left(right) ideals of groupoid-ordered groupoid. These results are generalizations of their results.

• Kyungpook Mathematical Journal
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• 제58권2호
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• pp.399-426
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• 2018

Cofinite graphs and cofinite groupoids are defined in a unified way extending the notion of cofinite group introduced by Hartley. These objects have in common an underlying structure of a directed graph endowed with a certain type of uniform structure, called a cofinite uniformity. Much of the theory of cofinite directed graphs turns out to be completely analogous to that of cofinite groups. For instance, the completion of a directed graph Γ with respect to a cofinite uniformity is a profinite directed graph and the cofinite structures on Γ determine and distinguish all the profinite directed graphs that contain Γ as a dense sub-directed graph. The completion of the underlying directed graph of a cofinite graph or cofinite groupoid is observed to often admit a natural structure of a profinite graph or profinite groupoid, respectively.

• East Asian mathematical journal
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• 제35권1호
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• pp.17-22
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• 2019

A travel groupoid is a pair (V, ${\ast}$) of a set V and a binary operation ${\ast}$ on V satisfying two axioms. For a travel groupoid, we can associate a graph in a certain manner. For a given graph G, we say that a travel groupoid (V, ${\ast}$) is on G if the graph associated with (V, ${\ast}$) is equal to G. There are some results on the classification of travel groupoids which are on a given graph [1, 2, 3, 9]. In this article, we introduce the notion of vertex-indexed partition systems on a graph, and classify the travel groupoids on the graph by the those vertex-indexed partition systems.

• Korean Journal of Mathematics
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• 제27권1호
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• pp.151-163
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• 2019

The aim of this paper is to consider the categorical equivalence between crossed modules within groupoids and 2-groupoids; and then relate normality and quotient in these two categories.

• International Journal of Fuzzy Logic and Intelligent Systems
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• 제7권1호
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• pp.7-16
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• 2007

First, we investigate fuzzy equivalence relations on a set X in the sense of Youssef and Dib. Second, we discuss fuzzy congruences generated by a given fuzzy relation on a fuzzy groupoid. In particular, we obtain the characterizations of ${\rho}\;o\;{\sigma}{\in}$ FC(S) for any two fuzzy congruences ${\rho}\;and\;{\sigma}$ on a fuzzy groupoid ($S,{\odot}$). Finally, we study the lattice of fuzzy equivalence relations (congruences) on a fuzzy semigroup and give certain lattice theoretic properties.