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For [Formula: see text], let [Formula: see text] be the semigroup of all singular mappings on [Formula: see text]. For each nonempty subset [Formula: see text] of [Formula: see text], let [Formula: see text] be the semigroup of all [Formula: see text]-decreasing mappings on [Formula: see text]. In this paper we determine the rank and idempotent rank of the semigroup [Formula: see text].

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Abstract. 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 mpotent 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) n n!. 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.