Short presentations for transformation monoids

TL;DR

This paper investigates minimal presentations of transformation monoids, establishing lower bounds on the number of additional relations needed beyond symmetric groups, and provides explicit presentations that resolve open problems.

Contribution

It determines the minimal number of additional relations required for presentations of full transformation, inverse, and partial transformation monoids, and offers explicit presentations for these monoids.

Findings

Minimum additional relations for $T_n$, $I_n$, and $PT_n$ are at least 4, 3, and 8 respectively.

Explicit presentations with specific numbers of relations are provided for various $n$.

Answers to open problems in the literature regarding monoid presentations are given.

Abstract

Due to the theorems of Cayley and Vagner-Preston, the full transformation monoids and the symmetric inverse monoids play analogous roles in the theory of monoids and inverse monoids, as the symmetric group does in the theory of groups. Every presentation for the finite full transformation monoids $T_n$, symmetric inverse monoids $I_n$, and partial transformation monoids $PT_n$ contains a monoid presentation for the symmetric group. In this paper we show that the number of relations required, in addition to those for the symmetric group, for each of these monoids are at least $4$, $3$, and $8$, respectively. We also give presentations for: $T_n$ with $4$ additional relations when $n$ is odd and $n\geq 5$; and $5$ additional relations for all $n\geq 4$; for $I_n$ with $3$ additional relations for all $n \geq 3$; and for $PT_n$ with $9$ relations for all $n\geq 4$. The presentations for $T_n$ and $I_n$ answer open problems in the literature.