Short presentations for transformation monoids

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 P_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 ≥ 5; and 5 additional relations for all n ≥ 4; for I_n with 3 additional relations for all n ≥ 3; and for PT_n with 9 relations for all n ≥ 4. The presentations for T_n and I_n answer open problems in the literature.