On deterministic finite automata and syntactic monoid size

We investigate the relationship between regular languages and syntactic monoid size. In particular, we consider the transformation monoids of n-state (minimal) deterministic finite automata. We show tight upper and lower bounds on the syntactic monoid size depending on the number of generators (input alphabet size) used. It turns out, that the two generator case is the most involved one. There we show a lower bound of n(n) (1 - 2/rootn) for the size of the syntactic monoid of a language accepted by an n-state deterministic finite automaton with binary input alphabet. Moreover, we prove that for every prime n greater than or equal to 7, the maximal size semigroup w.r.t. its size among all (transformation) semigroups which can be generated with two generators, is generated by a permutation with two cycles (of appropriate lengths) and a non-bijective mapping merging elements from these two cycles. As a by-product of our investigations we determine the maximal size among all semigroups generated by two transformations, where one is a permutation with a single cycle and the other is a non-bijective mapping. (C) 2004 Elsevier B.V. All rights reserved.