Locally-zero Groupoids and the Center of Bin(X)

TL;DR

This paper introduces the concept of the center of the semigroup of binary systems on a set and characterizes when a groupoid belongs to this center as being locally-zero.

Contribution

It defines the center of the semigroup of binary systems and characterizes groupoids in this center as locally-zero groupoids.

Findings

Groupoids in the center satisfy a specific set equality.

A groupoid is in the center if and only if it is locally-zero.

Abstract

In this paper we introduce the notion of the center $ZBin(X)$ in the semigroup $Bin(X)$ of all binary systems on a set $X$, and show that if $(X,\bullet)\in ZBin(X)$, then $x\not=y$ implies $\{x,y\}=\{x\bullet y,y\bullet x\}$.Moreover, we show that a groupoid $(X,\bullet )\in ZBin(X)$ if and only if it is a locally-zero groupoid.