SynopsisLet Singn be the subsemigroup of singular elements of the full transformation semigroup on a totally ordered finite set with n elements. Let be the subsemigroup of all decreasing maps of Singn. In this paper it is shown that is a non-regular abundant semigroup with n − 1 -classes and . Moreover, is idempotent-generated and it is generated by the n(n − 1)/2 idempotents in J*n−1. LetandSome recurrence relations satisfied by J*(n, r) and sh (n, r) are obtained. Further, it is shown that sh (n, r) is the complementary signless (or absolute) Stirling number of the first kind.

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Background: Pelvic inflammatory disease refers to any infection in the female lower reproductive tract that spreads to the upper reproductive tract. The disease comprises a spectrum of inflammatory disorders of the upper female genital tract, including any combination of endometritis, salpingitis, tubo-ovarian abscess and pelvic peritonitis.

SynopsisLet In be the symmetric inverse semigroup on Xn = {1,…, n}, let Sln be the subsemigroup of strictly partial one-to-one self-maps of Xn and let = { α ∊ SIn: x} ≦ x = U = ∅= be the semigroup of all partial one-to-one decreasing maps including the empty or zero map of Xn. In this paper it is shown that is an (irregular, for n ≧ 2) type A semigroup with n D*-classes and D* = I*. Further, it is shown that is generated by the n(n + l)/2 quasi-idempotents in

Let PO n be the semigroup of all order-preserving partial transformations of a finite chain. It is shown that |PO n | = c n satisfies the recurrence (2n − 1)(n + 1)c n+1 = 4 3n 2 − 1 c n − (2n + 1)(n − 1)c n−1 with initial conditions c 0 = 1, c 1 = 2. It is also shown that |E(PO n )| = e n satisfies the recurrence e n+1 = 5(e n − e n−1 ) + 1 with initial conditions e 0 = 1, e 1 = 2. Moreover, the cardinalities of the Green's relations L, R and J have been computed.