The Maximum Inverse Subsemigroup of a Near Permutation Semigroup#

^#Communicated by P. Higgins.

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 ± bℛ-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.

Keywords:

• Inverse semigroups
• Near permutation semigroups
• Transformation semigroups

Mathematics Subject Classification:

• Primary 20M20
• Secondary 20M18

ACKNOWLEDGMENTS

The author acknowledges the financial support of the Sub-Programa Ciência e Tecnologia do 2^o Quadro Comunitário de Apoio (grant number BD/18012/98), and project POCTI/32440/MAT/2000 (CAUL) of FCT and FEDER.

Notes

^#Communicated by P. Higgins.