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Dynamic effect algebras and their representations

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Abstract

For lattice effect algebras, the so-called tense operators were already introduced by Chajda and Kolařík. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that in every effect algebra can be introduced tense operators which, for non-complete lattice effect algebras, can be only partial mappings. An effect algebra equipped with tense operators reflects changes of quantum events from past to future. A crucial problem concerning tense operators is their representation. Having an effect algebra with tense operators, we can ask if there exists a frame such that each of these operators can be obtained by our construction. We solve this problem for (strict) dynamic effect algebras having a full set of homorphisms into a complete lattice effect algebra.

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Acknowledgments

We thank the anonymous referee for the very thorough reading and contributions to improve our presentation of the paper. I. Chajda and J. Paseka acknowledge the support by ESF Project CZ.1.07/2.3.00/20.0051 Algebraic methods in Quantum Logic of the Masaryk University. Supported by

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Authors and Affiliations

  1. Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Ivan Chajda

  2. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Jan Paseka

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  1. Ivan Chajda
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  2. Jan Paseka
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Correspondence to Jan Paseka.

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Chajda, I., Paseka, J. Dynamic effect algebras and their representations. Soft Comput 16, 1733–1741 (2012). https://doi.org/10.1007/s00500-012-0857-x

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  • DOI: https://doi.org/10.1007/s00500-012-0857-x

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