sylvester monoid

Cain, A. J., and A. Malheiro. "Deciding conjugacy in sylvester monoids and other homogeneous monoids." Int. J. Algebra Comput.. 25 (2015): 899-915. AbstractWebsite

We give a combinatorial characterization of conjugacy in the sylvester monoid, showing that conjugacy is decidable for this monoid. We then prove that conjugacy is undecidable in general for homogeneous monoids and even for multihomogeneous monoids.

Cain, A. J., R. D. Gray, and A. Malheiro. "Rewriting systems and biautomatic structures for Chinese, hypoplactic, and sylvester monoids." Int. J. Algebra Comput.. 25 (2015): 51-80. AbstractWebsite

This paper studies complete rewriting systems and biautomaticity for three interesting classes of finite-rank homogeneous monoids: Chinese monoids, hypoplactic monoids, and sylvester monoids. For Chinese monoids, we first give new presentations via finite complete rewriting systems, using more lucid constructions and proofs than those given independently by Chen & Qui and Güzel Karpuz; we then construct biautomatic structures. For hypoplactic monoids, we construct finite complete rewriting systems and biautomatic structures. For sylvester monoids, which are not finitely presented, we prove that the standard presentation is an infinite complete rewriting system, and construct biautomatic structures. Consequently, the monoid algebras corresponding to monoids of these classes are automaton algebras in the sense of Ufnarovskij.