The rank of the inverse semigroup of all partial automorphisms on a finite crown

Abstract

For $n\in \mathbb{N}$, let $[n]=\{1,2,\dots ,n\}$ be an $n$ - element set. As usual, we denote by $I_n$ the symmetric inverse semigroup on $[n]$, i.e. the partial one-to-one transformation semigroup on $[n]$ under composition of mappings. The crown (cycle) ${\mathcal{C}}_n$ is an $n$-ordered set with the partial order $\prec$ on $[n]$, where the only comparabilities are $$1 \prec 2 \succ 3 \prec 4 \succ \cdots \prec n \succ 1 ~~\mbox{ or }~~ 1 \succ 2 \prec 3 \succ 4 \prec \cdots \succ n \prec 1.$$ We say that a transformation $\alpha \in I_n$ is order-preserving if $x\prec y$ implies that $x\alpha \prec y\alpha$, for all $x,y$ from the domain of $\alpha$. In this paper, we study the inverse semigroup $I{C}_n$ of all partial automorphisms on a finite crown ${\mathcal{C}}_n$. We consider the elements, determine a generating set of minimal size and calculate the rank of $I{C}_n$.

References in corpus (1)

• On the semigroup of all partial fence-preserving injections on a finite set