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Idempotents in partial transformation semigroups

Synopsis An element α of Pn, the semigroup of all partial transformations of {1,2,…, n}, is said to have projection characteristic (r, s), or to belong to the set [r, s], if dom α= r, im α = s. Let E be the set of all idempotents in Pn\[n, n] and E1, the set of those idempotents with projection characteristic (n, n − 1) or (n − 1, n − 1). For α in Pn\[n, n], we define a number g(α), called the gravity of α and closely related to the number denned in Howie [5] for full transformations, and we obtain the result that Let d(α) be the defect of α, and for any real number x let [x] be the least integer m such that m ≧ x. Then by analogy with the results of Saito [9] we have that α ϵ Ek(α) and α ∉ Ek(α) where k(α) = [g(α)/d(α)] or [g(α)/d(α)+ 1. Following Howie, Lusk and McFadden [6] we then explore connections between the defect and the gravity of α. Letting K(n, r) be the subsemigroup of Pn consisting of all elements of rank r or less, we prove a result, corresponding to that of Howie and McFadden [7] for total transformations, that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n + 1, r + 1), the Stirling number of the second kind.