Subvarieties of the Variety Generated by the Five-element Brandt Semigroup

It is well-known that B2 is an inverse 0-simple semigroup that has played a distinguished role in the theory of semigroups. Let B2 be the variety generated by B2, and let B0 be the variety generated by the subsemigroup B0 = {0, d, cd, dc} of B2. Since all semigroups with five or fewer elements are finitely based [5], the variety B2 and its subvariety B0 are finitely based. In fact, it has been proved in [3] that every subvariety of B2 is finitely based. In this paper, standard techniques will be used to investigate the subvarieties of B2. It will be shown in Sec. 3 that the variety B0 contains a unique maximal subvariety. In Sec. 4, the subvarieties of B2 that do not contain any semilattices will be characterized, while in Sec. 5, those that contain semilattices will be characterized. The main result of this paper is a complete classification (Theorem 5.10) of the subvarieties of B2. In the final section, the finitely generated subvarieties of B2 will be investigated.