Yagci, Melek2025-03-232025-03-2320230126-67052180-4206https://doi.org/10.1007/s40840-022-01447-1https://hdl.handle.net/11486/6741For n is an element of N, let D-n be the semigroup of all order-decreasing transformations on X-n = {1, ... , n}, under its natural order, and let E(D-n) be the set of all idempotents in D-n and for any xi is an element of E(D-n), let D-n(xi) = {alpha e D-n : alpha(m) = xi for some m is an element of Z(+)}. In this paper, first we show that D-n(xi) is the maximum nilpotent subsemigroup of D-n with zero element xi. Moreover, we determine the minimum generating set of D-n(xi), and so the rank of D-n(xi). Also, we find the cardinality of D-n(xi).eninfo:eu-repo/semantics/closedAccessOrder-decreasing transformationMinimum generating setRankNilpotent subsemigroupsArticle46210.1007/s40840-022-01447-12-s2.0-85144485734Q1WOS:000901722100001Q1