Limit varieties of monoids satisfying a certain identity

A variety is finitely based if it has a finite basis of identities. A minimal non-finitely based variety is called limit. A monoid is aperiodic if all its subgoups are trivial. Limit varieties of aperiodic monoids have been studied by Jackson, Lee, Zhang and Luo, Gusev and Sapir. In particular, Gusev and Sapir have recently reduced the problem of classifying all limit varieties of aperiodic monoids to the two tasks. One of them is to classify limit varieties of monoids satisfying $xsxt approx xsxtx$. In this paper, we completely solve this task. In particular, we exhibit the first example of a limit variety of monoids with countably infinitely many subvarieties. In view of the result by Jackson and Lee, the smallest known monoid generating a variety with continuum many subvarieties is of order six. It follows from the result by Edmunds et al. that if there exists a smaller example, then up to isomorphism and anti-isomorphism, it must be a unique monoid $P_2^1$ of order five. Our main result implies that the variety generated by $P_2^1$ contains only finitely based subvarieties and so has only countably many of them.