The relative rank of the endomorphism monoid of a finite $G$-set

Abstract

For a group $G$ acting on a set $X$, let ${\text{End}}_G(X)$ be the monoid of all $G$-equivariant transformations, or $G$-endomorphisms, of $X$, and let ${\text{Aut}}_G(X)$ be its group of units. After discussing few basic results in a general setting, we focus on the case when $G$ and $X$ are both finite in order to determine the smallest cardinality of a set $W\subseteq {\text{End}}_G(X)$ such that $W\cup {\text{Aut}}_G(X)$ generates ${\text{End}}_G(X)$; this is known in semigroup theory as the relative rank of ${\text{End}}_G(X)$ modulo ${\text{Aut}}_G(X)$.

15 pages. To appear in Semigroup Forum

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• Cellular Automata and Finite Groups
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