On the size of inverse semigroups given by generators

Authors:
Martin Beaudry;Markus Holzer
Affiliations:
Département dinformatique, Université de Sherbrooke, 2500, boul. Université, Sherbrooke (Québec), J1K 2R1, Canada;Institut für Informatik, Universität Gieβen, Ardntstraβe 2, D-35392 Gieβen, Germany
Venue:
Theoretical Computer Science
Year:
2011

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Abstract

The size of the transformation semigroup of a reversible deterministic finite automaton with n states, or equivalently, of a semigroup given by generators of injective partial functions on n objects, had remained unexplored in the case where the set of generators is a pair. We show that in this case, the maximal size is attained by a semigroup generated by a permutation that satisfies a property depending on n and a partial injective mapping whose domain and image both have size n-1. Moreover, we give precise formulas in terms of n for this maximal size.