On sofic monoids

We investigate a notion of soficity for monoids. A group is sofic as a group if and only if it is sofic as a monoid. All finite monoids, all commutative monoids, all free monoids, all cancellative one-sided amenable monoids, all multiplicative monoids of matrices over a field, and all monoids obtained by adjoining an identity element to a semigroup are sofic. On the other hand, although the question of the existence of a non-sofic group remains open, we prove that the bicyclic monoid is not sofic. This shows that there exist finitely presented amenable inverse monoids that are non-sofic.     

The Myhill property for strongly irreducible subshifts over amenable groups

Let G be an amenable group and let A be a finite set. We prove that if X ? A ^G is a strongly irreducible subshift then X has the Myhill property, that is, every pre-injective cellular automaton ?? : X ?? X is surjective.