  We consider blind, deterministic, finite automata equipped with a register
which stores an element of a given monoid, and which is modified by right
multiplication by monoid elements. We show that, for monoids M drawn from a
large class including groups, such an automaton accepts the word problem of a
group H if and only if H has a finite index subgroup which embeds in the group
of units of M. In the case that M is a group, this answers a question of Elston
and Ostheimer.
