THE SEMIGROUPS OF BINARY SYSTEMS AND SOME PERSPECTIVES
Hee Sik Kim, J. Neggers · Bulletin of the Korean Mathematical Society · 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.