Characterization of Cross varieties of $J$-trivial monoids

A finitely based, finitely generated variety with finitely many subvarieties is a Cross variety. In the present article, it is shown that a variety of $J$-trivial monoids is Cross if and only if it excludes as subvarieties a certain list of 14 almost Cross varieties. Consequently, the list of 14 varieties exhausts all almost Cross varieties of $J$-trivial monoids.