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

Abstract

For $n \in \mathbb N$, let $[n] = \{1, 2, \ldots , n\}$ 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

$$\begin{aligned} 1 \prec 2 \succ 3 \prec 4 \succ \cdots \prec n \succ 1 ~~ \text{ or } ~~ 1 \succ 2 \prec 3 \succ 4 \prec \cdots \succ n \prec 1. \end{aligned}$$

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 $IC_n$ 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 $IC_n$.