Constructing finitely presented monoids which have no finite complete presentation (Q1357103)

A complete rewriting system \(R\) over an alphabet \(\Sigma\) consists of a set of rules \(u\to v\) for words from the word monoid \(\Sigma^*\) such that the relation \(\to\) is noetherian and confluent. An application of a rule \(u\to v\) leads from a word \(x=x_1ux_2\) to \(y=x_1vx_2\), denoted by \(x\to_Ry\). Let \(\to^*_R\) be the reflexive and transitive closure of \(\to_R\), and let \(\leftrightarrow^*_R\) be the least congruence on \(\Sigma^*\) that contains \(\to_R\). The main features of a complete rewriting system \(R\) are: (1) for any word \(x\in\Sigma^*\) there exists a unique irreducible element \(\widehat x\) (for which no rule is applicable) such that \(x\to^*\widehat x\); (2) \(x\leftrightarrow^*_Ry\) if and only if \(\widehat x=\widehat y\). A monoid \(M\) is said to be presented by \(R\) if \(M=\Sigma^*/\leftrightarrow_R^*\). \textit{C. C. Squier} [J. Pure Appl. Algebra 49, 201-217 (1987; Zbl 0648.20045)] showed that there are finitely presented monoids (and groups) that have a solvable word problem, but which are not presented by a complete rewriting system. The authors of the present paper show that there is such a monoid that has a presentation with trivial homotopy (and hence this monoid satisfies Squier's condition \(FP_\infty\)).