Coherent presentations for stylic monoids via row rewriting

Abstract

We study stylic monoids from the viewpoint of higher-dimensional rewriting. A coherent presentation records not only the defining relations of a monoid, but also the relations among them. Using the $N$-tableau model, we construct a finite coherent presentation for every stylic monoid of finite rank. We start from the known finite convergent row presentation. By adjoining one generating confluence for each critical branching, Squier's coherence theorem yields a finite coherent extension. We then simplify the resulted coherent presentation by two successive homotopical reductions adapted to the row combinatorics of $N$-tableaux. Finally, coherent Tietze transformations and a further homotopical reduction transfer the construction from the auxiliary row generators to the standard letter generators. This gives an effectively computable finite homotopy basis for the standard letter presentation. The construction is illustrated in detail in rank $2$.