Abstract
In some of their earlier papers the authors established correspondences between varieties of automata and semigroups and then found inconvenient the usual notion of transition semigroup of an automaton for dealing with automata satisfying some irregular identities. This is the motivation for assigning a new kind of semigroups to directable automata, which is done in this paper. The properties of these, so-called characteristic, semigroups are studied and semigroups that can be characteristic semigroups of directable automata are characterized. Finally, a correspondence between irregular varieties of automata and certain varieties of semigroups is given.
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