Identity bases for some non-exact varieties. (Q1882658)

The author studies the subvariety lattice of the variety \({\mathbf A}_2\) generated by the 5-element idem\-po\-tent-generated 0-simple semigroup \(A_2\). \textit{N. R. Reilly} (unpublished) has shown that \({\mathbf A}_2\) has a unique maximal subvariety denoted by \(\overline{\mathbf A}_2\) in the paper under review. The author finds finite identity bases for \(\overline{\mathbf A}_2\) (Theorem 2.7) and for its largest subvariety \(\overline{\mathbf B}_2\) that does not contain the 5-element Brandt semigroup \(B_2\) (Theorem 3.6). The intersection of \(\overline{\mathbf B}_2\) with the variety \({\mathbf B}_2\) generated by \(B_2\) turns out to be a unique maximal subvariety of \({\mathbf B}_2\). From this and from a result by \textit{Gy. Pollák} [Semigroup Forum 25, 9-23 (1982; Zbl 0496.20042)] the author deduces that every subvariety of \({\mathbf B}_2\) possesses a finite identity basis (Corollary 3.8).