- Communications Faculty of Sciences University Ankara Series A1 Mathematics and Statistics
- Vol: 71 Issue: 3
- Combinatorial results of collapse for order-preserving and order-decreasing transformations
Combinatorial results of collapse for order-preserving and order-decreasing transformations
Authors : Emrah Korkmaz
Pages : 769-777
Doi:10.31801/cfsuasmas.1019458
View : 9 | Download : 5
Publication Date : 2022-09-30
Article Type : Research
Abstract :The full transformation semigroup T n Tn is defined to consist of all functions from X n = { 1 , … , n } Xn={1,…,n} to itself, under the operation of composition. In \cite{JMH1}, for any α α in T n Tn , Howie defined and denoted collapse by c ( α ) = ⋃ t ∈ \im ( α ) { t α − 1 : | t α − 1 | ≥ 2 } c(α)=⋃t∈\im(α){tα−1:|tα−1|≥2} . Let O n On be the semigroup of all order-preserving transformations and C n Cn be the semigroup of all order-preserving and decreasing transformations on X n Xn= under its natural order, respectively. Let E ( O n ) E(On) be the set of all idempotent elements of O n On , E ( C n ) E(Cn) and N ( C n ) N(Cn) be the sets of all idempotent and nilpotent elements of C n Cn , respectively. Let U U be one of { C n , N ( C n ) , E ( C n ) , O n , E ( O n ) } {Cn,N(Cn),E(Cn),On,E(On)} . For α ∈ U α∈U , we consider the set \im c ( α ) = { t ∈ \im ( α ) : | t α − 1 | ≥ 2 } \imc(α)={t∈\im(α):|tα−1|≥2} . For positive integers 2 ≤ k ≤ r ≤ n 2≤k≤r≤n , we define U ( k ) = { α ∈ U : t ∈ \im c ( α ) and | t α − 1 | = k } , U ( k , r ) = { α ∈ U ( k ) : ∣ ∣ ⋃ t ∈ \im c ( α ) t α − 1 | = r } . 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 ) | |U(k,r)| , and so | U ( k ) | |U(k)| for some values r r and k k .Keywords : Order-preserving/decreasing transformation, collapse, nilpotent, idempotent