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Abstract. This work investigates the rank properties of A + (Bn), the additive semigroup reduct of affine near-semiring over Brandt semigroup Bn. In this connection, this work reports the ranks r 1 , r 2 , r 3 and r 5 of A + (Bn) and identifies a lower bound for the upper rank r 4 (A + (Bn)). While this lower bound is found to be the r 4 (A + (Bn)) for n ≥ 6, in other cases where 2 ≤ n ≤ 5, the upper rank of A + (Bn) is still open for investigation.

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We prove common fixed-point theorems in a metric space, which generalizes the result of Jay G. Mehta and M.L. Joshi, using (E.A)-property and Common Limit Range Property (CLR-property).

We describe and count the maximal subsemigroups of many well-known monoids of transformations and monoids of partitions. More precisely, we find the maximal subsemigroups of the full spectrum of monoids of order-or orientationpreserving transformations and partial permutations considered by V. H. Fernandes and co-authors (12 monoids in total); the partition, Brauer, Jones, and Motzkin monoids; and certain further monoids.Although descriptions of the maximal subsemigroups of some of the aforementioned classes of monoids appear in the literature, we present a unified framework for determining these maximal subsemigroups. This approach is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors. This allows us to concisely present the descriptions of the maximal subsemigroups, and to more clearly see their common features.