Extensions and submonoids of automatic monoids

AbstractIn this paper, submonoids and extensions of automatic and p-automatic monoids are studied. The concept of a p-automatic monoid is a variant on the usual concept of an automatic monoid designed to allow a geometric characterization analogous to the group case. In the case of right cancellative monoids, the two concepts coincide. Here, we study rational submonoids of (p-)automatic monoids, being able to show in many cases that (p-)automaticity is inherited. Our sharpest results concern rational subgroups. Also, closure properties are established for various notions of extensions of (p-)automatic monoids, including different types of products, ideal extensions, and Rees matrix constructions