Abstract
We provide new solutions to the word problems for the variety B2 generated by the five element Brandt semigroup B2 with zero divisors and the variety NB2 generated by B2 and the two element left and right zero semigroups. We also provide a finite basis of identities for the variety NB2. This leads to a complete description of the interval [B2, NB2], in the lattice of semigroup varieties. This is a critical interval in the study of the lattice of aperiodic Rees-Sushkevich varieties. In addition, the varieties B2 and NB2 are especially important in the characterization of those Rees-Sushkevich varieties that are exact, that is, generated by completely 0-simple semigroups. Byproducts of this note are techniques that lend themselves to a deeper study of the structure of the free objects in the varieties in the interval [B2, NB2].
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