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Galois connections and tense operators on q-effect algebras. (English) Zbl 1387.03071
It is known that tense operators on a Boolean algebra can be obtained by the canonical construction from a time frame and that this is not true for effect algebras in general. The authors [Soft Comput. 16, No. 10, 1733–1741 (2012; Zbl 1318.03059)] found special conditions that ensures this representation. They introduce a q-effect algebra (an effect algebra with two specific unary operations $$d$$ and $$q$$ derived from the theory of MV-algebras), q-tense operators (using a Galois connection preserving $$d$$ and $$q$$) and q-states. The representation is constructed for q-tense operators on some q-effect algebras (such that every q-state is Jauch-Piron and the set of q-states is order reflecting).
MSC:
 03G12 Quantum logic 06A15 Galois correspondences, closure operators (in relation to ordered sets) 06C15 Complemented lattices, orthocomplemented lattices and posets 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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