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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. |
