Consider the finite set X n = {1,2, …, n } ordered in the standard way. Let T n denote the full transformation semigroup on X n , that is, the semigroup of all mappings α: X n → X n under composition. We shall call α order-preserving if i ≤ j implies i α ≤ j α for i , j ∈ X n , and α is decreasing if i α ≤ i for all i ∈ X n . This paper investigates combinatorial properties of the semigroup O of all order-preserving mappings on X n , and of its subsemigroup C which consists of all decreasing and order-preserving mappings.
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Peter M. Higgins (1993) studied this question.