The semigroups of linearly realizable finite automata. I

Abstract

First the problem is solved how one can decide whether an arbitrary finite semigroup H is linearly A-realizable, i.e., whether there exists a linearly realizable finite automaton having a semigroup isomorphic to H. This leads to a question about the existence of certain generating subsets of H. The determination of these subsets is rather complicated in case H-HH=O and very simple in case H-HH#O. But in the first case we are able to clear up completely the structure of the semigroups which are linearly A-realizable: These are exactly the finite right groups which have maximal subgroups of the type described by Ecker in [4]. In the second case we get only necessary structure conditions. Among other things we shall see: If a semigroup H is linearly A-realizable one can define a congruence relation ? on it having the property, that H is isomorphic to a semigroup of a strongly connected and linearly realizable automaton iff the so-called index of H equals the index of H/?. Developing these results about semigroups we obtain at the same time many structure theorems about linearly realizable automata.