The maximum inverse subsemigroup of a near permutation semigroup

A transformation semigroup over a set X with N elements is said to be a near permutation semigroup if it is generated by a group G of permutations on N elements and by a set H of transformations of rank N - 1. For near permutation semigroups S = ≪ G, H ≫, where H is a group, we consider a group of permutations, whose elements are constructed from the elements of H . Without loss of generality, we identify the identity of H to the idempotent. The condition 2 O G ( S ) (1), where, is a necessary condition for S to be inverse and is a sufficient one for S to be b -unipotent. We characterize the subsemilattices and the maximal subsemilattices of the near permutation semigroups satisfying the above condition. With those characterizations of a semilattice E contained in a semigroup S, we determine the maximum inverse subsemigroup of S which has E as its subsemilattice of idempotents. We use this result to test whether a near permutation semigroup is inverse.