  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.
