Green's relations and unit-regularity for semigroup of transformations whose characters are bijective

Let $X$ be a nonempty set and $\mathcal{P}=\{X_i\colon i\in I\}$ be a partition of $X$. Denote by $T(X, \mathcal{P})$ the semigroup of all transformations of $X$ that preserve $\mathcal{P}$. In this paper, we study the semigroup $\mathcal{B}(X,\mathcal{P})$ of all transformations $f\in T(X, \mathcal{P})$ such that $\chi^{(f)}\in {\rm Sym}(I)$, where ${\rm Sym}(I)$ is the symmetric group on $I$ and $\chi^{(f)}\colon I \to I$ is the character (map) of $f$ defined by $i\chi^{(f)}=j$ whenever $X_if\subseteq X_j$. We describe 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 describe Green's relations on $\mathcal{B}(X,\mathcal{P})$, and prove that $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$ when $\mathcal{P}$ is finite. We also give a necessary and sufficient condition for $\mathcal{D} = \mathcal{J}$ on $\mathcal{B}(X,\mathcal{P})$. We end the paper with a conjecture.