Semitransitive subsemigroups of the singular part of the finite symmetric inverse semigroup

TL;DR

This paper determines the smallest size of semitransitive subsemigroups within the singular part of the finite symmetric inverse semigroup and classifies all such minimal examples.

Contribution

It establishes the minimal cardinality of these subsemigroups and provides a complete classification of all minimal cases.

Findings

Minimal cardinality is 2n - p + 1, where p is the largest proper divisor of n.

All semitransitive subsemigroups of minimal size are classified.

The structure of these subsemigroups is fully characterized.

Abstract

We prove that the minimal cardinality of the semitransitive subsemigroup in the singular part $\IS_n\setminus \S_n$ of the symmetric inverse semigroup $\IS_n$ is $2n-p+1$, where $p$ is the greatest proper divisor of $n$, and classify all semitransitive subsemigroups of this minimal cardinality.