Short presentations for transformation monoids
By 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 groups do 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 is at least $4$, $3$, and $8$, respectively. We also give presentations for: $T_n$ with $4$ additional relations when $n\geq 7$; for $I_n$ with $3$ additional relations for all $n \geq 3$; and for $PT_n$ with $8$ additional relations for all $n\geq 7$. The presentations for $T_n$ and $I_n$ answer open problems in the literature.