Cayley automaton semigroups

Abstract

Let S be a semigroup, C(S) the automaton constructed from the right Cayley graph of S with respect to all of S as the generating set and ∑(C(S)) the automaton semigroup constructed from C(S). Such semigroups are termed Cayley automaton semigroups. For a given semigroup S we aim to establish connections between S and ∑(C(S)). For a finite monogenic semigroup S with a non-trivial cyclic subgroup C[sub]n we show that ∑(C(S)) is a small extension of a free semigroup of rank n, and that in the case of a trivial subgroup ∑(C(S)) is finite. The notion of invariance is considered and we examine those semigroups S satisfying S ≅ ∑(C(S)). We classify which bands satisfy this, showing that they are those bands with faithful left-regular representations, but exhibit examples outwith this classification. In doing so we answer an open problem of Cain. Following this, we consider iterations of the construction and show that for any n there exists a semigroup where we can iterate the construction n times before reaching a semigroup satisfying S ≅ ∑(C(S)). We also give an example of a semigroup where repeated iteration never produces a semigroup satisfying S ≅ ∑(C(S)). Cayley automaton semigroups of infinite semigroups are also considered and we generalise and extend a result of Silva and Steinberg to cancellative semigroups. We also construct the Cayley automaton semigroup of the bicyclic monoid, showing in particular that it is not finitely generated.