Partial transformation monoids preserving a uniform partition

TL;DR

This paper investigates the structure of the monoid of partial transformations preserving a uniform partition, providing a presentation, generators, and order, thereby deepening understanding of its algebraic properties.

Contribution

It introduces a detailed algebraic analysis of the monoid, including its generation, order, and presentation, which was not previously established.

Findings

Monoid is a quotient of a wreath product.

It is generated by 5 elements.

The order of the monoid is computed.

Abstract

The objective of this paper is to study the monoid of all partial transformations of a finite set that preserve a uniform partition. In addition to proving that this monoid is a quotient of a wreath product with respect to a congruence relation, we show that it is generated by 5 generators, we compute its order and determine a presentation on a minimal generating set.