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Algebraic Duality

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Book cover Duality Theories for Boolean Algebras with Operators

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Abstract

In this chapter, the algebraic duality that exists between relational structures and complete and atomic Boolean algebras with operators is studied. Every relational structure corresponds to a uniquely determined dual complete and atomic Boolean algebra with operators, namely the complex algebra of the structure, and conversely, every complete and atomic Boolean algebra with operators corresponds to a uniquely determined dual relational structure, namely the atom structure of the algebra. The duality between structures and algebras carries with it a corresponding duality between morphisms: every bounded homomorphism between relational structures corresponds to a dual complete homomorphism between the dual algebras, and conversely. The duality between the morphisms implies other dualities as well. For example, every inner subuniverse of a relational structure corresponds to a complete ideal in the dual algebra, and vice versa. Every bounded congruence on a relational structure corresponds to a complete subuniverse of the dual algebra, and vice versa. The disjoint union of a system of relational structures corresponds to the direct product of the system of dual algebras, and vice versa.

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  1. Department of Mathematics, Mills College, Oakland, CA, USA

    Steven Givant

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  1. Steven Givant
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Givant, S. (2014). Algebraic Duality. In: Duality Theories for Boolean Algebras with Operators. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06743-8_1

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