Definition:Center (Abstract Algebra)/Semigroup

This page is about Center in the context of Semigroup. For other uses, see Center.

Definition

The center of a semigroup $\struct {S, \circ}$, denoted $\map Z S$, is the subset of elements in $S$ that commute with every element in $G$.

Symbolically:

$\map Z S = \set {s \in S: \forall x \in S: s \circ x = x \circ s}$

Historical Note

The notation $\map Z S$, conventionally used for the center of a structure $\struct {S, \circ}$, derives from the German Zentrum, meaning center.

Linguistic Note

The British English spelling of center is centre.

The convention on $\mathsf{Pr} \infty \mathsf{fWiki}$ is to use the American English spelling center, but it is appreciated that there may be lapses.