The semigroups of binary systems and some perspectives. (English) Zbl 1172.20047

Kim, Hee Sik; Neggers, Joseph

Bull. Korean Math. Soc. 45, No. 4, 651-661 (2008).

Summary: Given binary operations “\(*\)” and “\(\circ\)” on a set \(X\), define a product binary operation “\(\square\)” as follows: \(x\square y:= (x*y)\circ(y*x)\). This in turn yields a binary operation on \(\text{Bin}(X)\), the set of groupoids defined on \(X\) turning it into a semigroup \((\text{Bin}(X),\square)\) 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.

MSC:

20N02	Sets with a single binary operation (groupoids)
08A02	Relational systems, laws of composition
20M99	Semigroups

Keywords:

semigroups of binary systems; leftoids; groupoids; orbits