Ranks of nilpotent subsemigroups of order-preserving and decreasing transformation semigroups

Let Cn be the semigroup of all order-preserving and decreasing transformations on X = {1, . . . , n} under its natural order, and let N(Cn) be the subsemigroup of all nilpotent elements of Cn . For 1 ≤ r ≤ n − 1, let N(Cn,r) = {α ∈ N(Cn) : |im (α)| ≤ r}, Nr(Cn) = {α ∈ N(Cn) : α is an m-potent for any 1 ≤ m ≤ r}. In this paper we find the cardinality and the rank of the subsemigroup N(Cn,r) of Cn . Moreover, we show that the set Nr(Cn) is a subsemigroup of N(Cn) and then, we find a lower bound for the rank of Nr(Cn) .

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