Largest 2-generated subsemigroups of the symmetric inverse semigroup

Proceedings of the Edinburgh Mathematical Society, nº50 (2007), p.551-561

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, if M(n) denotes this maximum, it is shown that M(n)/|In| → 1 as n → ∞. Furthermore, we may deduce, the already known fact, that In embeds as a local submonoid of an inverse 2-generated subsemigroup of In+1.