  We show that for any monoid M, the family of languages accepted by M-automata
(or equivalently, generated by regular valence grammars over M) is completely
determined by that part of M which lies outside the maximal ideal. Hence, every
such family arises as the family of languages accepted by N-automata where N is
a simple or 0-simple monoid. A consequence is that every such family is either
the class of regular languages, contains all the blind one-counter languages,
or is the family of languages accepted by G-automata for G a non-locally-finite
torsion group.
  We consider a natural extension of the usual definition which permits the
automata to utilise more of the structure of each monoid, and also allows us to
define S-automata for S an arbitrary semigroup. In the monoid case, the
resulting automata are equivalent to the valence automata with rational target
sets} which arise in the theory of regulated rewriting systems. We study the
case that the register semigroup is completely simple or completely 0-simple,
obtaining a complete characterisation of the classes of languages corresponding
to such semigroups in terms of their maximal subgroups. In the process, we
obtain a number of results about rational subsets of Rees matrix semigroups
which may be of independent interest.
