6,069 research outputs found

    The distributivity spectrum of Baker's variety

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    For every nn, we evaluate the smallest kk such that the congruence inclusion α(β∘nγ)⊆αβ∘kαγ\alpha (\beta \circ_n \gamma ) \subseteq \alpha \beta \circ_{k} \alpha \gamma holds in a variety of reducts of lattices introduced by K. Baker. We also study varieties with a near-unanimity term and discuss identities dealing with reflexive and admissible relations.Comment: v3, v4 improvements and simplifications v5, v6 minor fixe

    The Logic of Partitions: Introduction to the Dual of the Logic of Subsets

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    Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms--which is reflected in the duality between quotient objects and subobjects throughout algebra. Modern categorical logic as well as the Kripke models of intuitionistic logic suggest that the interpretation of ordinary "propositional" logic might be the logic of subsets of a given universe set. If "propositional" logic is thus seen as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic

    Quantum logic is undecidable

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    We investigate the first-order theory of closed subspaces of complex Hilbert spaces in the signature (∨,⊥,0,1)(\lor,\perp,0,1), where `⊥\perp' is the orthogonality relation. Our main result is that already its quasi-identities are undecidable: there is no algorithm to decide whether an implication between equations and orthogonality relations implies another equation. This is a corollary of a recent result of Slofstra in combinatorial group theory. It follows upon reinterpreting that result in terms of the hypergraph approach to quantum contextuality, for which it constitutes a proof of the inverse sandwich conjecture. It can also be interpreted as stating that a certain quantum satisfiability problem is undecidable.Comment: 11 pages. v3: improved exposition. v4: minor clarification
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