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Profinite algebras are the residually finite compact algebras, whose underlying topological spaces are Stone spaces. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at Stone spaces as the dual spaces of Boolean algebras, we find a simple characterization of when the dual space admits a natural structure of topological algebra. This provides a new approach to duality theory which, in the case of a Stone signature, culminates in the proof that a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for residually closed Stone varieties is also established.
We investigate computable metrizability of Polish spaces up to homeomorphism. In this paper we focus on Stone spaces. We use Stone duality to construct the first known example of a computable topological Polish space not homeomorphic to any computabl
We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean algebras with
Under a general categorical procedure for the extension of dual equivalences as presented in this papers predecessor, a new algebraically defined category is established that is dually equivalent to the category $bf LKHaus$ of locally compact Hausdor
Assuming Jensons principle diamond: Whenever B is a totally imperfect set of real numbers, there is special Aronszajn tree with no continuous order preserving map into B.
We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in descriptive set theoretic (topological) sense and in measure theoretic sense.