  Let $\mathbf{A}$ be a finite simple non-abelian Mal'cev algebra (e.g. a
group, loop, ring). We investigate the Boolean power $\mathbf{D}$ of
$\mathbf{A}$ by the countable atomless Boolean algebra $\mathbf{B}$ filtered at
some idempotents $e_1,\dots,e_n$ of $\mathbf{A}$. When $e_1,\dots,e_n$ are all
idempotents of $\mathbf{A}$ we establish two concrete representations of
$\mathbf{D}$: as the Fra\"iss\'e limit of the class of finite direct powers of
$\mathbf{A}$, and as congruence classes of the countable free algebra in the
variety generated by $\mathbf{A}$. Further, for arbitrary $e_1,\dots,e_n$, we
show that $\mathbf{D}$ is $\omega$-categorical and that its automorphism group
has the small index property, strong uncountable cofinality and the Bergman
property. As necessary background we establish some general properties of
congruences and automorphisms of filtered Boolean powers of $\mathbf{A}$ by any
Boolean algebra $\mathbf{B}$, including a semidirect decomposition for their
automorphism groups.
