Combinatorial results of collapse for order-preserving and order-decreasing transformations

Authors : Emrah KORKMAZ

Pages : 769-777

Doi:10.31801/cfsuasmas.1019458

View : 49 | Download : 12

Publication Date : 2022-09-30

Article Type : Research Paper

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 insert ignore into journalissuearticles values( α ); = ⋃ t ∈ \im insert ignore into journalissuearticles values( α ); { t α − 1 : | t α − 1 | ≥ 2 } cinsert ignore into journalissuearticles values(α);=⋃t∈\iminsert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values( O n ); Einsert ignore into journalissuearticles values(On); be the set of all idempotent elements of O n On , E insert ignore into journalissuearticles values( C n ); Einsert ignore into journalissuearticles values(Cn); and N insert ignore into journalissuearticles values( C n ); Ninsert ignore into journalissuearticles values(Cn); be the sets of all idempotent and nilpotent elements of C n Cn , respectively. Let U U be one of { C n , N insert ignore into journalissuearticles values( C n ); , E insert ignore into journalissuearticles values( C n ); , O n , E insert ignore into journalissuearticles values( O n ); } {Cn,Ninsert ignore into journalissuearticles values(Cn);,Einsert ignore into journalissuearticles values(Cn);,On,Einsert ignore into journalissuearticles values(On);} . For α ∈ U α∈U , we consider the set \im c insert ignore into journalissuearticles values( α ); = { t ∈ \im insert ignore into journalissuearticles values( α ); : | t α − 1 | ≥ 2 } \imcinsert ignore into journalissuearticles values(α);={t∈\iminsert ignore into journalissuearticles values(α);:|tα−1|≥2} . For positive integers 2 ≤ k ≤ r ≤ n 2≤k≤r≤n , we define U insert ignore into journalissuearticles values( k ); = { α ∈ U : t ∈ \im c insert ignore into journalissuearticles values( α );   and   | t α − 1 | = k } , U insert ignore into journalissuearticles values( k , r ); = { α ∈ U insert ignore into journalissuearticles values( k ); : ∣ ∣ ⋃ t ∈ \im c insert ignore into journalissuearticles values( α ); t α − 1 | = r } . Uinsert ignore into journalissuearticles values(k);={α∈U:t∈\imcinsert ignore into journalissuearticles values(α); and |tα−1|=k},Uinsert ignore into journalissuearticles values(k,r);={α∈Uinsert ignore into journalissuearticles values(k);:|⋃t∈\imcinsert ignore into journalissuearticles values(α);tα−1|=r}. The main objective of this paper is to determine | U insert ignore into journalissuearticles values( k , r ); | |Uinsert ignore into journalissuearticles values(k,r);| , and so | U insert ignore into journalissuearticles values( k ); | |Uinsert ignore into journalissuearticles values(k);| for some values r r and k k .
Keywords : Order preserving decreasing transformation, collapse, nilpotent, idempotent

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