Combinatorial results of collapse for order-preserving and order-decreasing transformations
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Date
2022
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Ankara Üniversitesi
Abstract
The full transformation semigroup Tn is defined to consist of all functions from Xn={1,…,n} to itself, under the operation of composition. In \cite{JMH1}, for any α in Tn, Howie defined and denoted collapse by c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2}. Let On be the semigroup of all order-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 α∈U, we consider the set \imc(α)={t∈\im(α):|tα−1|≥2}. For positive integers 2≤k≤r≤n, we define U(k)={α∈U:t∈\imc(α) and |tα−1|=k},U(k,r)={α∈U(k):∣∣⋃t∈\imc(α)tα−1|=r}. The main objective of this paper is to determine |U(k,r)|, and so |U(k)| for some values r and k.
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Keywords
Order-preserving/decreasing transformation, collapse, nilpotent, idempotent