Regularity of some invariant transformation semigroups (Q6650688)

For a non-empty set \(X\) and for a nonempty subset \(Y\) of \(X\), let \(T(X)\) denote the full transformation semigroup on \(X\) (under composition) and let \[T_{Y}(X)=\{ \alpha \in T(X) : Y\alpha\subseteq Y\},\] which is a subsemigroup of \(T(X)\). In this paper, the authors define the following subsemigroups and characterize the regularity of them: \N\begin{align*} \NM_{Y}(X) &= \{ \alpha \in T_{Y}(X) : \alpha \mbox{ is one to one} \},\\\NE_{Y}(X) &= \{ \alpha \in T_{Y}(X) : \alpha \mbox{ is onto} \},\\\NAM_{Y}(X)&= \{ \alpha \in T_{Y}(X) : \{ x\in X : \lvert x\alpha\alpha^{-1}\rvert >1 \} \mbox{ is finite}\},\\\NAE_{Y}(X)&= \{ \alpha \in T_{Y}(X) : X \setminus \mathrm{ran\,}(\alpha) \mbox{ is finite}\},\\\NOM_{Y}(X)&= T_{Y}(X)\setminus AM_{Y}(X) \mbox{ and}\\\NOE_{Y}(X)&= T_{Y}(X)\setminus AE_{Y}(X). \N\end{align*} \NFor an infinite \(X\), it is shown that every element in \(M_{Y}(X)\) is regular in \(T_{Y}(X)\) if and only if \(Y=X\) or \(Y\) is finite, and that every element in \(E_{Y}(X)\) is regular in \(T_{Y}(X)\) if and only if \(Y=X\) or \(\lvert Y\rvert =1\) or \(X\setminus Y \) is finite.\N\NFor any nonempty set \(X\), other main results of this paper are listed below:\N\begin{itemize}\N\item[\((i)\)] \(M_{Y}(X)\) is regular if and only if \(X\) is finite;\N\item[\((ii)\)] \(E_{Y}(X)\) is regular if and only if \(X\) is finite\N\item[\((iii)\)] \(AM_{Y}(X)\) is regular if and only if \(X\) is finite and \(Y=X\) or \(\lvert Y\rvert =1\);\N\item[\((iv)\)] \(AE_{Y}(X)\) is regular if and only if \(X\) is finite and \(Y=X\) or \(\lvert Y\rvert =1\);\N\item[\((v)\)] \(AM_{Y}(X)\cap AE_{Y}(X)\) is regular if and only if \(Y=X\) or \(\lvert Y\rvert =1\);\N\item[\((vi)\)] \(OM_{Y}(X)\) and \(OE_{Y}(X)\) are non-regular semigroups; and\N\item[\((vii)\)] \(OM_{Y}(X)\cap OE_{Y}(X)\) is regular if and only if \(Y=X\) or \(\lvert Y\rvert =1\).\N\end{itemize}