# Random ubiquitous transformation semigroups

## TL;DR

This paper investigates the properties of transformation semigroups, introducing the concept of ubiquity, and shows that randomly generated semigroups often satisfy conditions leading to efficient, irredundant small generating sets.

## Contribution

It provides a sufficient condition for a transformation semigroup to be ubiquitous and demonstrates that random generation typically produces semigroups meeting this condition.

## Key findings

- Randomly generated transformation semigroups asymptotically satisfy the ubiquity condition.
- The naive algorithm's output is irredundant under the ubiquity condition.
- Ubiquity ensures minimal generating sets are of the same size, aiding efficient semigroup analysis.

## Abstract

A smallest generating set of a semigroup is a generating set of the smallest cardinality. Similarly, an irredundant generating set $X$ is a generating set such that no proper subset of $X$ is also a generating set. A semigroup $S$ is ubiquitous if every irredundant generating set of $S$ is of the same cardinality. We are motivated by a na\"{i}ve algorithm to find a small generating set for a semigroup, which in practice often outputs a smallest generating set. We give a sufficient condition for a transformation semigroup to be ubiquitous and show that a transformation semigroup generated by $k$ randomly chosen transformations asymptoticly satisfies the sufficient condition. Finally, we show that under this condition the output of the previously mentioned na\"{i}ve algorithm is irredundant.