Akademik Veri Yönetim Sistemi
COMMUNICATIONS FACULTY OF SCIENCES UNIVERSITY OF ANKARA-SERIES A1 MATHEMATICS AND STATISTICS, cilt.71, sa.3, ss.769-777, 2022 (ESCI)
The full transformation semigroup Tn is defined to consist of all functions from Xn = {1, ... , n} to itself, under the operation of com-position. In [9], for any alpha in Tn, Howie defined and denoted collapse by c(alpha) = U t & ISIN;im(alpha) preserving transformations and Cn be the semigroup of all order-preserving and decreasing transformations on Xn under its natural order, respectively. Let E(On) be the set of all idempotent elements of On, E(Cn) and N(Cn) be the sets of all idempotent and nilpotent elements of Cn, respectively. Let U be one of {Cn, N(Cn), E(Cn), On, E(On)}. For alpha & ISIN; U, we consider the set imc(alpha) = {t & ISIN; im(alpha) : |t alpha-1| & GE; 2}. For positive integers 2 & LE; k & LE; r & LE; n, we define {t alpha-1 : |t alpha-1| & GE; 2}. Let On be the semigroup of all order-U(k) = {alpha & ISIN; U : t & ISIN; imc(alpha) and |t alpha-1|= k}, ?? ? U(k, r) = {alpha & ISIN; U(k) : t alpha-1| = r}. t & ISIN;imc(alpha) The main objective of this paper is to determine |U(k, r)|, and so |U(k)| for some values r and k.