Limit varieties of $J$-trivial monoids

We show that limit varieties of monoids recently discovered by Gusev, Zhang and Luo and their subvarieties are generated by monoids of the form $M_τ(W)$ for certain congruences $τ$ on the free monoid. The construction $M_τ(W)$ is a generalization of widely used Dilworth-Perkins construction. Using this construction, we find explicit generators for Gusev limit varieties and give a short reproof to the fact that Zhang-Luo limit variety is non-finitely based.

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