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Order related concepts for arbitrary groupoids
김희식,Joseph Neggers,소금숙 대한수학회 2017 대한수학회보 Vol.54 No.4
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.
Semi-neutral groupoids and $BCK$-algebras
김희식,Joseph Neggers,서영주 대한수학회 2022 대한수학회논문집 Vol.37 No.3
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.
THE SEMIGROUPS OF BINARY SYSTEMS AND SOME PERSPECTIVES
Kim, Hee-Sik,Neggers, Joseph Korean Mathematical Society 2008 대한수학회보 Vol.45 No.4
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.
The semigroups of binary systems and some perspectives
Hee Sik Kim,Joseph Neggers 대한수학회 2008 대한수학회보 Vol.45 No.4
Given binary operations “*” and “◦” on a set X, define a product binary operation “∗” as follows: x∗y := (x * y) ◦ (y * 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), ∗)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 ∗ 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. Given binary operations “*” and “◦” on a set X, define a product binary operation “∗” as follows: x∗y := (x * y) ◦ (y * 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), ∗)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 ∗ 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.
ORDER RELATED CONCEPTS FOR ARBITRARY GROUPOIDS
Kim, Hee Sik,Neggers, Joseph,So, Keum Sook Korean Mathematical Society 2017 대한수학회보 Vol.54 No.4
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.