• The Pure and Applied Mathematics
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• v.19 no.2
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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 $C^*({\Gamma})$.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.217-227
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• 2024

Let G be an infinite countable group and A be a finite set. If Σ ⊆ A^G is a strongly irreducible subshift of finite type and 𝓖 is the local conjugacy equivalence relation on Σ. We construct a decreasing sequence 𝓡 of unital C^*-subalgebras of C(Σ) and a sequence of faithful conditional expectations E defined on C(Σ), and obtain a Toeplitz algebra 𝓣 (𝓡, 𝓔) and a C^*-algebra C^*(𝓡, 𝓔) for the pair (𝓡, 𝓔). We show that C^*(𝓡, 𝓔) is *-isomorphic to the reduced groupoid C^*-algebra C^*_r(𝓖).

• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.601-631
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• 2010

In [6] and [7], we introduced graph von Neumann algebras which are the (groupoid) crossed product algebras of von Neumann algebras and graph groupoids via groupoid actions. We showed that such crossed product algebras have the graph-depending amalgamated reduced free probabilistic properties. In this paper, we will consider a scalar-valued $W^*$-probability on a given graph von Neumann algebra. We show that a diagonal graph $W^*$-probability space (as a scalar-valued $W^*$-probability space) and a graph W¤-probability space (as an amalgamated $W^*$-probability space) are compatible. By this compatibility, we can find the relation between amalgamated free distributions and scalar-valued free distributions on a graph von Neumann algebra. Under this compatibility, we observe the scalar-valued freeness on a graph von Neumann algebra.

• Korean Journal of Mathematics
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• v.32 no.3
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• pp.533-544
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• 2024

Given a groupoid (X, *) and a real-valued function d : X → R, a new (derived) function Φ(X, *)(d) is defined as [Φ(X, *)(d)](x, y) := d(x * y) + d(y * x) and thus Φ(X, *) : R^X → R^X2 as well, where R is the set of real numbers. The mapping Φ(X, *) is an R-linear transformation also. Properties of groupoids (X, *), functions d : X → R, and linear transformations Φ(X, *) interact in interesting ways as explored in this paper. Because of the great number of such possible interactions the results obtained are of necessity limited. Nevertheless, interesting results are obtained. E.g., if (X, *, 0) is a groupoid such that x * y = 0 = y * x if and only if x = y, which includes the class of all d/BCK-algebras, then (X, *) is *-metrizable, i.e., Φ(X, *)(d) : X² → X is a metric on X for some d : X → R.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.375-381
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• 2019

In this paper, we introduce an operation denoted by [$Br_e$], a bracket operation, which maps an arbitrary groupoid ($X,{\ast}$) on a set X to another groupoid $(X,{\bullet})=[Br_e](X,{\ast})$ which on groups corresponds to sending a pair of elements (x, y) of X to its commutator $xyx^{-1}y^{-1}$. When applied to classes such as d-algebras, BCK-algebras, a variety of results is obtained indicating that this construction is more generally useful than merely for groups where it is of fundamental importance.

• Korean Journal of Mathematics
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• v.26 no.2
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• pp.175-189
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• 2018

In this paper, at first we generalize the notion of algebra over a field. A ${\Gamma}$-algebra is an algebraic structure consisting of a vector space V, a groupoid ${\Gamma}$ together with a map from $V{\times}{\Gamma}{\times}V$ to V. Then, on every associative ${\Gamma}$-algebra V and for every ${\alpha}{{\in}}{\Gamma}$ we construct an ${\alpha}$-Lie algebra. Also, we discuss some properties about ${\Gamma}$-Lie algebras when V and ${\Gamma}$ are the sets of $m{\times}n$ and $n{\times}m$ matrices over a field F respectively. Finally, we define the notions of ${\alpha}$-derivation, ${\alpha}$-representation, ${\alpha}$-nilpotency and prove Engel theorem in this case.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.649-658
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• 2022

In this paper, we introduce the notion of a left-almost-zero groupoid, and we generalize two axioms which play important roles in the theory of BCK-algebra using the notion of a projection. Moreover, we investigate a Smarandache disjointness of semi-leftoids.

• Bulletin of the Korean Mathematical Society
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• v.45 no.4
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• pp.651-661
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• 2008

Given binary operations "*" and "$\circ$" on a set X, define a product binary operation "$\Box$" as follows: $x{\Box}y\;:=\;(x\;{\ast}\;y)\;{\circ}\;(y\;{\ast}\;x)$. This in turn yields a binary operation on Bin(X), the set of groupoids defined on X turning it into a semigroup (Bin(X), $\Box$)with identity (x * y = x) the left zero semigroup and an analog of negative one in the right zero semigroup (x * y = y). The composition $\Box$ is a generalization of the composition of functions, modelled here as leftoids (x * y = f(x)), permitting one to study the dynamics of binary systems as well as a variety of other perspectives also of interest.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1373-1386
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• 2017

In this paper, we introduce and explore suggested notions of 'above', 'below' and 'between' in general groupoids, Bin(X), as well as in more detail in several well-known classes of groupoids, including groups, semigroups, selective groupoids (digraphs), d/BCK-algebras, linear groupoids over fields and special cases, in order to illustrate the usefulness of these ideas. Additionally, for groupoid-classes (e.g., BCK-algebras) where these notions have already been accepted in a standard form, we look at connections between the several definitions which result from our introduction of these ideas as presented in this paper.

• Kyungpook Mathematical Journal
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• v.61 no.2
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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.