On the ranks of semigroups of transformations on a finite set with restricted range

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Let $\mathcal{PT}(X)$ be the semigroup of all partial transformations on $X$ (under composition) and let $\mathcal{T}(X)$ and $\mathcal{I}(X)$ be the subsemigroups of $\mathcal{PT}(X)$ of all full transformations on $X$ and of all injective partial transformations on $X$, respectively. Given a nonempty subset $Y$ of $X$, let $\mathcal{PT}(X,Y)=\{\alpha\in\mathcal{PT}(X)\mid X\alpha\subseteq Y\}$, $\mathcal{T}(X,Y)=\mathcal{PT}(X,Y)\cap\mathcal{T}(X)$ and $\mathcal{I}(X,Y)=\mathcal{PT}(X,Y)\cap\mathcal{I}(X)$. In 2008, Sanwong and Sommanee described the largest regular subsemigroup and determined the Green's relations of $\mathcal{T}(X,Y)$. In this paper, we present analogous results for both $\mathcal{PT}(X,Y)$ and $\mathcal{I}(X,Y)$. For a finite set $X$ such that $|X|\ge3$, the ranks of $\mathcal{PT}(X)=\mathcal{PT}(X,X)$, $\mathcal{T}(X)=\mathcal{T}(X,X)$ and $\mathcal{I}(X)=\mathcal{I}(X,X)$ are well known to be $4$, $3$ and $3$, respectively. In this paper, we also compute the ranks of $\mathcal{PT}(X,Y)$, $\mathcal{T}(X,Y)$ and $\mathcal{I}(X,Y)$, for any proper nonempty subset $Y$ of $X$.
Original languageUnknown
Pages (from-to)497-510
JournalAlgebra Colloquium
Issue number3
Publication statusPublished - 1 Jan 2014

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