Elements of finite order for finite monadic Church-Rosser Thue systems

Abstract

A Thue system T T over Σ \Sigma is said to allow nontrivial elements of finite order, if there exist a word u ∈ Σ ∗ u \in {\Sigma ^ \ast } and integers n ≥ 0 n \ge 0 and k ≥ 1 k \ge 1 such that u ↮ T ∗ λ u \nleftrightarrow \,_T^ \ast \lambda and u n + k ↔ T ∗ u n {u^{n + k}} \leftrightarrow \,_T^ \ast {u^n} . Here the following decision problem is shown to be decidable: Instance. A finite, monadic, Church-Rosser Thue system T T over Σ \Sigma . Question. Does T T allow nontrivial elements of finite order? By a result of Muller and Schupp this implies in particular that given a finite monadic Church-Rosser Thue system T T it is decidable whether the monoid presented by T T is a free group or not.