Largest 2-generated subsemigroups of the symmetric inverse semigroup

Abstract The symmetric inverse monoid In is the set of all partial permutations of an n-element set. The largest possible size of a 2-generated subsemigroup of In is determined. Examples of semigroups with these sizes are given. Consequently, ifM(n) denotes this maximum, it is shown thatM(n)/|In | → 1 as n→∞. Furthermore, we may deduce, the already known fact, that In embeds as a local submonoi