Inverse near permutation semigroups

by Jorge André

A transformation semigroup $S=<G,{\cal U}>$ over a set with $N$ elements is said to be a near permutation semigroup if $G$ is a group of permutations on $N$ elements and ${\cal U}$ is a set of transformations of $rank\; N-1$. In this talk we give necessary and sufficient conditions for a near permutation semigroup $S=<G,H>$, where $H$ is a group, to be inverse. Moreover, we obtain conditions which guarantee, that its semilattice of idempotents is generated by the idempotents of $S$ of $rank$ greater than $N-2$ or $rank$ greater than $N-3$.