Solution of the membership problem of the prefix monoid in certain one-relator groups

Abstract

In this work we examine to what extent small cancellation conditions imply the solution of the membership problem of the prefix monoid in one-relator groups. The main results show that with additional combinatorial conditions on the defining relator (prefix pureness and strongly cyclically reduceness), which synchronize the prefixes and an analogue of the Greendlinger Lemma for the set of prefixes (Property P(\(n\))), the membership problem for the prefix monoid can be solved in cases not covered by previous results.