On the (m, r)-potent ranks of certain semigroups of transformations
Ping Zhao, Taijie You, Yunkun Chen · Journal of Algebra and Its Applications · 2015
It is known that the ranks of the semigroups 𝒮ingnand 𝒮𝒫𝒯n(the semigroups of singular self-maps and partial transformations on Xn= {1, 2, …, n}, respectively) are [Formula: see text], respectively. The idempotent ranks, defined as the smallest number of idempotents generating set, of 𝒮ingnand 𝒮𝒫𝒯nare the same value as the ranks, respectively. Idempotent can be seen as a special case (with m = r = 1) of (m, r)-potent. In this paper, we determine the (m, r)-potent ranks, defined as the smallest number of (m, r)-potent generating set, of the semigroups 𝒮ingnand 𝒮𝒫𝒯n.