• 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.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.161-166
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• 2005

Let BQ be the class of all semigroups whose bi-ideals are quasi-ideals. It is known that regular semigroups, right [left] 0-simple semigroups and right [left] 0-simple semigroups belong to BQ. Every zero semigroup is clearly a member of this class. In this paper, we characterize when generalized full transformation semigroups and generalized Baer-Levi semigroups are in BQ in terms of the cardinalities of sets.

• Kyungpook Mathematical Journal
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• v.57 no.1
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• pp.99-108
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• 2017

In this paper we construct quadratic ${\beta}-algebras$ on a field, and we discuss both linear-quadratic ${\beta}-algebras$ and quadratic-linear ${\beta}-algebras$ in a field. Moreover, we discuss some relations of binary operations in ${\beta}-algebras$.