Formal languages and the irreducible word problem in groups
There exist structural classifications of groups with a regular, one-counter or context-free word problem. Following on from this, the main object of the work presented here has been the irreducible word problem of a group, a notion introduced by Haring-Smith, who denned it as the subset of the word problem consisting of the non-empty words which have no non-empty proper subword equal to the identity. He proved that the groups with a finite irreducible word problem with respect to some group generating set are exactly the plain groups.;We know that the class of groups with a context-free irreducible word problem is a proper subclass of the virtually free groups. We look at direct products of finitely generated free groups by finite groups and also at the plain groups and consider their irreducible word problems with respect to minimal group generating sets. We prove that, of all the direct products of the infinite cyclic group by a non-trivial finite group, only Z x Z2 and Z x Z 3 have context-free irreducible word problem (and only with respect to a few group generating sets). We also exhibit a plain group that has context-free irreducible word problem with respect to every minimal group generating set.;Looking at the direct products of finitely generated free groups by non-trivial finite groups, we have found that the only irreducible word problem that is one-counter is that of Z x Z2 with respect to the canonical group generating set.;As for irreducible word problems lying in classes of languages above context-free, on one hand, we prove that having a recursive irreducible word problem is equivalent to having a recursive word problem. On the other hand, we prove that, while there are groups such that the fact that their irreducible word problem is recursively enumerable implies that they are recursive, that is not always the case.