Limit varieties of aperiodic monoids

A limit variety is a variety that is minimal with respect to being non-finitely based. We present a new limit variety of aperiodic monoid. We also show that if there exists any other limit variety of aperiodic monoids, then it is contained in the joint of the variety $\mathbb B^1$ of all idempotent monoids and certain finitely generated variety $\mathbb E^1$ with $\mathbb B^1 \wedge \mathbb E^1 = \mathbb L_2^1$, where $\mathbb L_2^1$ is the variety of left-zero monoids. Jackson and Lee proved that $\mathbb E^1$ is HFB, that is, its every subvariety is finitely based. We exend this result a step up the classical decomposition $\mathbb B^1=\bigcup_{i \ge 2} \mathbb L^1_i$ by showing that $\mathbb E^1 \vee \overline{\mathbb E^1} \vee \mathbb L^1_3$ is also HFB, where $\overline{\mathbb E^1}$ is the variety dual of $\mathbb E^1$.