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