Quasi-automatic semigroups
Abstract
Quasi-automatic semigroups are extensions of a Cayley graph of an automatic group. Of course, a quasi-automatic semigroup generalizes an automatic semigroup. We observe that a semigroup S may be automatic only when S is finitely generated, while a semigroup may be quasi-automatic but it is not necessary finitely generated. Similar to the usual automatic semigroups, a quasi-automatic semigroup is closed under direct and free products. Furthermore, a semigroup S is graph automatic if and only if S with a zero element adjoined is graph automatic, and also a semigroup S is graph automatic if and only if S with an identity element adjoined is graph automatic. However, the class of quasi-automatic semigroups is a much wider class than the class of automatic semigroups. In this paper, we show that every automatic semigroup is quasi-automatic but the converse statement is not true (see Example ??). In addition, we notice that the quasi-automatic semigroups are invariant under the changing of generators, while a semigroup may be automatic with respect to a finite generating set but not the other. Finally, the connection between the quasi-automaticity of two semigroups S and T, where T is a subsemigroup with finite Rees index in S will be investigated and considered.
Communicated by A. R. Rajan
