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

TL;DR

This paper investigates the inverse semigroup of all partial automorphisms on a finite crown, determining its elements, minimal generating set, and rank, thus advancing understanding of its algebraic structure.

Contribution

It introduces the inverse semigroup of partial automorphisms on a finite crown and computes its rank, a novel contribution to the algebraic theory of inverse semigroups.

Findings

Identified all elements of the inverse semigroup

Determined a minimal generating set

Calculated the rank of the semigroup

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) $\cal{C}_n$ 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 $IC_n$ of all partial automorphisms on a finite crown $\cal{C}_n$. We consider the elements, determine a generating set of minimal size and calculate the rank of $IC_n$.