The Ranks of the Additive Semigroup Reduct of Affine Near-Semiring over Brandt Semigroup

This work investigates the rank properties of $A^{+}(B_n)$, the additive semigroup reduct of affine near-semiring over Brandt semigroup $B_n$. In this connection, this work reports the ranks $r_1$, $r_2$, $r_3$ and $r_5$ of $A^{+}(B_n)$ and identifies a lower bound for the upper rank $r_4(A^{+}(B_n))$. While this lower bound is found to be the $r_4(A^{+}(B_n))$ for $n\ge 6$, in other cases where $2\le n\le 5$, the upper rank of $A^{+}(B_n)$ is still open for investigation.