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

Abstract

The full transformation semigroup $T_n$ is defined to consist of all functions from $X_n=\{1,\dots ,n\}$ to itself, under the operation of composition. In \cite{JMH1}, for any $\alpha$ in $T_n$, Howie defined and denoted collapse by $c\left(\alpha \right)=\bigcup _{t\in \text{im}\left(\alpha \right)}\{t{\alpha }^{-1}:|t{\alpha }^{-1}|\ge 2\}$. Let $O_n$ be the semigroup of all order-preserving transformations and $C_n$ be the semigroup of all order-preserving and decreasing transformations on $X_n$under its natural order, respectively. Let $E\left(O_n\right)$ be the set of all idempotent elements of $O_n$, $E\left(C_n\right)$ and $N\left(C_n\right)$ be the sets of all idempotent and nilpotent elements of $C_n$, respectively. Let $U$ be one of $\{C_n,N(C_n),E(C_n),O_n,E(O_n\left)\right\}$. For $\alpha \in U$, we consider the set ${\text{im}}^c\left(\alpha \right)=\{t\in \text{im}(\alpha ):|t{\alpha }^{-1}|\ge 2\}$. For positive integers $2\le k\le r\le n$, we define $$\begin{array}{ccc}U\left(k\right)& =& \{\alpha \in U:t\in {\text{im}}^c(\alpha \left)\text{and}\right|t{\alpha }^{-1}|=k\},\\ U(k,r)& =& \{\alpha \in U(k):|\bigcup _{t\in {\text{im}}^c\left(\alpha \right)}t{\alpha }^{-1}|=r\}.\end{array}$$ The main objective of this paper is to determine $\left|U\right(k,r\left)\right|$, and so $\left|U\right(k\left)\right|$ for some values $r$ and $k$.

Keywords

• Order-preserving/decreasing transformation
• collapse
• nilpotent
• idempotent

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