THE SEMIGROUPS OF BINARY SYSTEMS AND SOME PERSPECTIVES

Given binary operations "*" and "circle" on a set X, define a product binary operation "square" as follows: x square y := (x * y) circle (y * x). This in turn yields a binary operation on (Bin(X), square)the set of groupoids defined on X turning it into a semigroup (Bin(X), square)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 square 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.