On Monoids Presented by Rewriting Systems and Automatic Structures for Their Submonoids

Generalizing results of Otto & Ruškuc, this paper shows that every finitely generated submonoid of a monoid presented by a confluent finite special rewriting system admits an automatic structure that is simultaneously rr-, `r-, r`-, and ``-automatic; and that every finitely generated submonoid of a monoid presented by a confluent regular monadic rewriting system admits an automatic structure that is simultaneously rrand ``-automatic. An algorithm is given to decide whether the monoid presented by a confluent monadic finite rewriting system is `ror r`-automatic. Should such a monoid be `ror r`-automatic, the rewriting system’s language of normal form words forms part of such an automatic structure. These results are applied to yield answers to some hitherto open questions and to recover and generalize established results.