Rank properties of the semigroup reducts of affine near-semirings over Brandt semigroups
There are several different ways to define ranks of semigroups based on the idea of independent subsets and generating subsets. \textit{J. M. Howie} and \textit{M. I. Marques Ribeiro} considered five such ranks in two papers [Commun. Algebra 27, No. 11, 5333--5347 (1999; Zbl 0940.20056); Southeast Asian Bull. Math. 24, No. 2, 231--237 (2000; Zbl 0967.20030)] denoted as \(r_{i}\), \(i=1,2,3,4,5\). In general, for any finite semigroup \(\Gamma\), \(r_{1}(\Gamma)\leq r_{2}(\Gamma)\leq r_{3}(\Gamma)\leq r_{4}(\Gamma)\leq r_{5}(\Gamma)\), and so they are called by small rank, lower rank, intermediate rank, upper rank and large rank of \(\Gamma\), respectively. For some semigroups, these five ranks are equal. Yet, for some other semigroups, these may be all distinct. In this article under review, the rank properties of the semigroup reducts of the affine near-semiring \(A^{+}(B_{n})\) over the Brandt semigroup \(B_{n}\) are investigated. Either the ranks are explicitly computed, or some bounds for the ranks are given.
scientific article; zbMATH DE number 6688587