LARGEST 2-GENERATED SUBSEMIGROUPS OF THE SYMMETRIC INVERSE SEMIGROUP

Abstract

The symmetric inverse monoid $\mathcal{I}_{n}$ is the set of all partial permutations of an $n$-element set. The largest possible size of a $2$-generated subsemigroup of $\mathcal{I}_{n}$ is determined. Examples of semigroups with these sizes are given. Consequently, if $M(n)$ denotes this maximum, it is shown that $M(n)/|\mathcal{I}_{n}|\rightarrow1$ as $n\rightarrow\infty$. Furthermore, we deduce the known fact that $\mathcal{I}_{n}$ embeds as a local submonoid of an inverse $2$-generated subsemigroup of $\mathcal{I}_{n+1}$.