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 $$A^+(B_n)$$ over the Brandt semigroup $$B_n$$ . We determine the small, lower, intermediate and large ranks of the additive semigroup reduct $$A_n$$ , and find a lower bound for the upper rank of $$A_n$$ . In case $$n \ge 6$$ , 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 $$M_n$$ , and provide lower bounds for the intermediate and upper ranks of $$M_n$$ .