Finite basis problem for 2-testable monoids

A monoid S ¹ obtained by adjoining a unit element to a 2-testable semigroup S is said to be 2-testable. It is shown that a 2-testable monoid S ¹ is either inherently non-finitely based or hereditarily finitely based, depending on whether or not the variety generated by the semigroup S contains the Brandt semigroup of order five. Consequently, it is decidable in quadratic time if a finite 2-testable monoid is finitely based.