Limit varieties generated by finite non-J-trivial aperiodic monoids

Jackson and Lee proved that certain six-element monoid generates a hereditarily finitely based variety \(\mathbb E^1\) whose lattice of subvarieties contains an infinite ascending chain. We identify syntactic monoids which generate finitely generated subvarieties of \(\mathbb E^1\) and show that one of these finite monoids together with certain seven-element monoid generates a new limit variety.