Let $CABA$ be the category of complete and atomic boolean algebras and complete boolean homomorphisms, and let $CSL$ be the category of complete meet-semilattices and complete meet-homomorphisms. We show that the forgetful functor from $CABA$ to $CSL$ has a left adjoint. This allows us to describe an endofunctor $H$ on $CABA$ such that the category $Alg(H)$ of algebras for $H$ is dually equivalent to the category $Coalg(\mathcal{P})$ of coalgebras for the powerset endofunctor $\mathcal{P}$ on $Set$. As a consequence, we derive Thomason duality from Tarski duality.

翻译：让美元CABA美元成为完整和原子布林代数和完整的布林同族体的类别,让美元CSL美元成为完整的美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式西西西西式西式西式西式西式西式西式西式西式西式西式西式西式西西西式西式西西西西西西西西西西西西式西式西西西西西式西西西西西西西西式西式西式西西西式西式西式西式美式西式西式西式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式美式西西西西西