The semigroups of binary systems and some perspectives

Bull. Korean Math. Soc. 2008 Vol. 45, No. 4, 651-661 Printed December 1, 2008

Hee Sik Kim and Joseph Neggers Hanyang University

Abstract : Given binary operations ``$*$" and ``$\circ$" on a set $X$, define a product binary operation ``$\Box$" as follows: $x\Box y:= (x*y)\circ (y*x)$. This in turn yields a binary operation on ${\rm Bin}(X)$, the set of groupoids defined on $X$ turning it into a semigroup \bn 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.