Rank properties of the semigroup reducts of affine near-semirings over Brandt semigroups

Abstract

We investigate the rank properties of the semigroup reducts of the affine near-semiring A+ (Bn) over the Brandt semigroup Bn.We determine the small, lower, intermediate and large ranks of the additive semigroup reduct An, and find a lower bound for the upper rank of An. In case n ≥ 6, we show that the lower bound is actually equal to the upper rank. We also find the small, lower, and large ranks of the multiplicative semigroup reduct Mn, and provide lower bounds for the intermediate and upper ranks of Mn.