On the Rank and the k-Idempotent Rank of the Semigroup CSPn,k
Let Pn and Sn be partial transformation semigroup and symmetry group on a finite chain Xn , respectively, if natural number n≥3. Let (Ck =〈gk〉) be a k-locally cyclic group on Xn , and let (CSPn,k=Ck∪(Pn\Sn)), if for any positive integer k such that 3≤k ≤n. It is easy to prove that CSPn,k is a subsemigroup of the partial transformation semigroup Pn. Through an analysis of the Green's relation and the idempotent of the semigroup CSPn,k ,the minimal generating set and the minimal generating set of k-idempotent are obtained. Further, the rank and the k-idempotent rank of the semigroup CSPn,k is further confirmed.
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