Abstract

Groups with automatic structure (automatic groups for short) have recently received a lot of attention in the literature. They have been shown to have many nice properties, and consequently it has been asked whether these results can be carried over to monoids with automatic structure. Here we show that there exist finitely presented monoids with automatic structure that cannot be presented through finite and convergent string-rewriting systems, thus answering a question in the negative that is still open for the class of automatic groups. Secondly, we present an automatic monoid that has an exponential Dehn function, which establishes another difference to the class of automatic groups. In fact, both our example monoids are bi-automatic.

Citations