On the Semigroup of partition-preserving transformations whose characters are bijective
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Let $\mathcal{P}=\{X_i\colon i\in I\}$ be a partition of a set $X$. We say that a transformation $f\colon X \to X$ preserves $\mathcal{P}$ if for every $X_i \in \mathcal{P}$, there exists $X_j \in \mathcal{P}$ such that $X_if \subseteq X_j$. Consider the semigroup $\mathcal{B}(X,\mathcal{P})$ of all transformations $f$ of $X$ such that $f$ preserves $\mathcal{P}$ and the character (map) $\chi^{(f)}\colon I \to I$ defined by $i\chi^{(f)}=j$ whenever $X_if\subseteq X_j$ is bijective. We describe Green's relations on $\mathcal{B}(X,\mathcal{P})$, and prove that $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$ if $\mathcal{P}$ is finite. We give a necessary and sufficient condition for $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$. We characterize unit-regular elements in $\mathcal{B}(X,\mathcal{P})$, and determine when $\mathcal{B}(X,\mathcal{P})$ is a unit-regular semigroup. We alternatively prove that $\mathcal{B}(X,\mathcal{P})$ is a regular semigroup. We end the paper with a conjecture.
Abstract
Let $\mathcal{P}=\{X_i\colon i\in I\}$ be a partition of a set $X$. We say that a transformation $f\colon X \to X$ preserves $\mathcal{P}$ if for every $X_i \in \mathcal{P}$, there exists $X_j \in \mathcal{P}$ such that $X_if \subseteq X_j$. Consider the semigroup $\mathcal{B}(X,\mathcal{P})$ of all transformations $f$ of $X$ such that $f$ preserves $\mathcal{P}$ and the character (map) $\chi^{(f)}\colon I \to I$ defined by $i\chi^{(f)}=j$ whenever $X_if\subseteq X_j$ is bijective. We describe Green's relations on $\mathcal{B}(X,\mathcal{P})$, and prove that $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$ if $\mathcal{P}$ is finite. We give a necessary and sufficient condition for $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$. We characterize unit-regular elements in $\mathcal{B}(X,\mathcal{P})$, and determine when $\mathcal{B}(X,\mathcal{P})$ is a unit-regular semigroup. We alternatively prove that $\mathcal{B}(X,\mathcal{P})$ is a regular semigroup. We end the paper with a conjecture.
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