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Logical Connectives on Lattice Effect Algebras

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Abstract

An effect algebra is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. In this article we present an approach to the study of lattice effect algebras (LEAs) that emphasizes their structure as algebraic models for the semantics of (possibly) non-standard symbolic logics. This is accomplished by focusing on the interplay among conjunction, implication, and negation connectives on LEAs, where the conjunction and implication connectives are related by a residuation law. Special cases of LEAs are MV-algebras and orthomodular lattices. The main result of the paper is a characterization of LEAs in terms of so-called Sasaki algebras. Also, we compare and contrast LEAs, Hájek’s BL-algebras, and the basic algebras of Chajda, Halaš, and Kühr.

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Authors and Affiliations

  1. Emeritus Professor of Mathematics, University of Massachusetts Amherst, Postal Address 1 Sutton Court, Amherst, MA, 01002, USA

    D. J. Foulis

  2. Mathematical Institute, Slovak Academy of Sciences, Stefánikova 49, SK-814 73, Bratislava, Slovakia

    S. Pulmannová

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  1. D. J. Foulis
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  2. S. Pulmannová
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Correspondence to S. Pulmannová.

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Foulis, D.J., Pulmannová, S. Logical Connectives on Lattice Effect Algebras. Stud Logica 100, 1291–1315 (2012). https://doi.org/10.1007/s11225-012-9454-3

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