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Dynamic effect algebras and their representations. (English) Zbl 1318.03059
Summary: For lattice effect algebras, the so-called tense operators were already introduced by I. Chajda and M. Kolařík [Math. Slovaca 62, No. 3, 379–388 (2012; Zbl 1324.03026)]. 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.

MSC:
03G12 Quantum logic
03G25 Other algebras related to logic
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