Maximal Specht varieties of monoids

A variety of algebras is a Specht variety if all its subvarieties are finitely based. This article presents the first example of a maximal Specht variety of monoids. The existence of such an example is counterintuitive since it is long known that maximal Specht varieties of semigroups do not exist. This example permits a characterization of Specht varieties in the following four classes based on identities that they must satisfy and varieties that they cannot contain: (1) overcommutative varieties, (2) varieties containing a certain monoid of order seven, (3) varieties of aperiodic monoids with central idempotents, and (4) subvarieties of the variety generated by the Brandt monoid of order six. Other results, including the uniqueness or nonexistence of limit varieties within the aforementioned four classes, are also deduced. Specifically, overcommutative limit varieties of monoids do not exist. In contrast, the limit variety of semigroups, discovered by M. V. Volkov in the 1980s, is an overcommutative variety.