# zbMATH — the first resource for mathematics

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)
Full Text:
##### References:
 [1] Botur, M.; Paseka, J., On tense MV-algebras, Fuzzy Sets Syst., 259, 111-125, (2015) · Zbl 1335.03069 [2] Burges, J., Basic tense logic, (Gabbay, D. M.; Günther, F., Handbook of Philosophical Logic, vol. II, (1984), D. Reidel Publ. Comp), 89-139 [3] Chajda, I., Algebraic axiomatization of tense intuitionistic logic, Cent. Eur. J. Math., 9, 1185-1191, (2011) · Zbl 1260.03113 [4] Chajda, I.; Janda, J.; Paseka, J., How to produce tense S-operators in lattice effect algebras, Found. Phys., 44, 792-811, (2014) · Zbl 1319.81013 [5] Chajda, I.; Kolařík, M., Dynamic effect algebras, Math. Slovaca, 62, 379-388, (2012) · Zbl 1324.03026 [6] Chajda, I.; Paseka, J., Dynamic effect algebras and their representation, Soft Comput., 16, 1733-1741, (2012) · Zbl 1318.03059 [7] Chajda, I.; Paseka, J., Tense operators and dynamic De Morgan algebras, (Proc. 2013 IEEE 43rd Internat. Symp. Multiple-Valued Logic, (2013), Springer), 219-224 [8] Cignoli, R. L.O.; D’Ottaviano, I. M.L.; Mundici, D., Algebraic foundations of many-valued reasoning, (2000), Kluwer · Zbl 0937.06009 [9] Diaconescu, D.; Georgescu, G., Tense operators on MV-algebras and łukasiewicz-moisil algebras, Fundam. Inform., 81, 379-408, (2007) · Zbl 1136.03045 [10] Dvurečenskij, A.; Pulmannová, S., New trends in quantum structures, (2000), Kluwer Academic Publishers, Ister Sci. Dordrecht/Boston/London, Bratislava · Zbl 0987.81005 [11] Dvurečenskij, A., Perfect effect algebras are categorically equivalent with abelian interpolation po-groups, J. Aust. Math. Soc., 82, 183-207, (2007) · Zbl 1117.06009 [12] Ewald, W. B., Intuitionistic tense and modal logic, J. Symb. Log., 51, 166-179, (1986) · Zbl 0618.03004 [13] Foulis, D. J.; Bennett, M. K., Effect algebras and unsharp quantum logics, Found. Phys., 24, 1325-1346, (1994) · Zbl 1213.06004 [14] Niederle, J.; Paseka, J., Homogeneous orthocomplete effect algebras are covered by MV-algebras, Fuzzy Sets Syst., 210, 89-101, (2013) · Zbl 1268.06009 [15] Paseka, J.; Janda, J., A dynamic effect algebras with dual operation, Math. Appl., 1, 79-89, (2012) · Zbl 1296.03039 [16] Paseka, J., Operators on MV-algebras and their representations, Fuzzy Sets Syst., 232, 62-73, (2013) · Zbl 1314.06016 [17] Pták, P.; Pulmannová, S., Orthomodular structures as quantum logics, (1991), Kluwer Academic Publishers Dordrecht/Boston/London · Zbl 0743.03039 [18] Riečanová, Z., Basic decomposition of elements and Jauch-piron effect algebras, Fuzzy Sets Syst., 155, 138-149, (2005) · Zbl 1073.81014 [19] Teheux, B., Algebraic approach to modal extensions of łukasiewicz logics, (2009), Université de Liege, doctoral thesis [20] Wijesekera, D., Constructive modal logics I, Ann. Pure Appl. Log., 50, 271-301, (1990) · Zbl 0714.03016
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.