ON THE m-POTENT RANKS OF CERTAIN SEMIGROUPS OF ORIENTATION PRESERVING TRANSFORMATIONS

It is known that the ranks of the semigroups SOPn, SPOPn and SSPOPn (the semigroups of orientation preserving singular selfmaps, partial and strictly partial transformations on Xn = {1, 2, . . . , n}, respectively) are n, 2n and n + 1, respectively. The idempotent rank, defined as the smallest number of idempotent generating set, of SOPn and SSPOPn are the same value as the rank, respectively. Idempotent can be seen as a special case (with m = 1) of m-potent. In this paper, we investigate the m-potent ranks, defined as the smallest number of m- potent generating set, of the semigroups SOPn, SPOPn and SSPOPn. Firstly, we characterize the structure of the minimal generating sets of SOPn. As applications, we obtain that the number of distinct minimal generating sets is (n−1)nn!. Secondly, we show that, for 1 ≤ m ≤ n−1, the m-potent ranks of the semigroups SOPn and SPOPn are also n and 2n, respectively. Finally, we find that the 2-potent rank of SSPOPn is n + 1.