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How to produce S-tense operators on lattice effect algebras. (English) Zbl 1319.81013
The paper continues the study of tense operators on MV-algebras, published in [M. Botur and J. Paseka, “On tense MV-algebras”, Fuzzy Sets Syst. (2014); {doi:10.1016/j.fss.2014.06.006}] and [J. Paseka, Fuzzy Sets Syst. 232, 62–73 (2013; Zbl 1314.06016)] Here the results are extended to effect algebras.
Effect algebras offer a very general representation of quantum events, using a partial operation $$+$$ (disjunction of mutually excluding events) and orthosupplement (generalized negation) $$'$$. In lattice effect algebras, we may use total operation $$x\oplus y=x+(y\land x')$$ and its dual, $$x\odot y=(x'\oplus y')'$$.
Tense operators admit to introduce time and dynamics and express logical statements like “it is always going to be the case that ...” and “it has always been the case that ...”. It is shown how tense operators can be constructed for a given time frame. The authors study also the inverse task, called a representation theorem: for given tense operators on an effect algebra, to find the respective time frame. The authors distinguish E-tense operators, which preserve $$\oplus,\odot$$ just as E-states, and S-tense operators $$G$$, which, moreover, are interior operators, i.e., $$G(x)\leq x$$, $$G(G(x))=x$$.
A state is a finitely additive probability measure. If a state $$s$$ on a lattice effect algebra satisfies $$s(x\oplus x)=s(x)\oplus s(x)$$, resp. $$s(x)=1=s(y) \implies s(x\land y)=1$$ , it is called an E-state, resp. a Jauch-Piron state. The E-states are extremal states, thus they do not form a convex set.
Further, E-semi states are studied; these are only subadditive and Jauch-Piron E-semi states are infima of sets of E-states.
If an S-tense effect algebra $$E$$ has an order reflecting (=order determining) set of E-states and all E-states are Jauch-Piron states, then $$E$$ can be embedded into a tense MV-algebra. A representation theorem for such lattice effect algebras with E-tense operators is proved and demonstrated on an example.

##### MSC:
 81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) 06C15 Complemented lattices, orthocomplemented lattices and posets 03G25 Other algebras related to logic 06A11 Algebraic aspects of posets
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##### References:
 [1] Belluce, LP; Grigolia, R; Lettieri, A, Representations of monadic MV-algebras, Stud. Log., 81, 123-144, (2005) · Zbl 1093.06008 [2] Botur, M., Paseka, J.: On tense MV-algebras. Fuzzy Set Syst (2014). doi:10.1016/j.fss.2014.06.006 · Zbl 1335.03069 [3] Burges, J.: Basic tense logic. In: Gabbay, D.M., Günther, F. (eds.) Handbook of Philosophical Logic, vol. II, pp. 89-139. D. Reidel Publ. Comp. (1984) · Zbl 1136.03045 [4] Butnariu, D., Klement, E.P.: Triangular Norm-Based Measures and Games with Fuzzy Coalitions. Kluwer Academic Publishers, Dordrecht (1993) · Zbl 0804.90145 [5] Cattaneo, G; Ciucci, D; Dubois, D, Algebraic models of deviant modal operators based on De Morgan and Kleene lattices, Inf. Sci., 181, 4075-4100, (2011) · Zbl 1242.03088 [6] Chajda, I; Kolařík, M, Dynamic effect algebras, Math. Slovaca, 62, 379-388, (2012) · Zbl 1324.03026 [7] Chajda, I; Paseka, J, Dynamic effect algebras and their representations, Soft Comput., 16, 1733-1741, (2012) · Zbl 1318.03059 [8] Chajda, I., Paseka, J.: Tense operators and dynamic De Morgan algebras, In: Proceedings of 2013 IEEE 43rd International Symposium Multiple-Valued Logic, pp. 219-224. Springer, Berlin (2013) · Zbl 1073.81014 [9] Cignoli, R.L.O., D’ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer, Dordrecht (2000) · Zbl 0937.06009 [10] Diaconescu, D; Georgescu, G, Tense operators on MV-algebras and łukasiewicz-moisil algebras, Fundam. Inf., 81, 379-408, (2007) · Zbl 1136.03045 [11] Nola, A; Navara, M, The $$σ$$-complete MV-algebras which have enough states, Colloq. Math., 103, 121-130, (2005) · Zbl 1081.06011 [12] Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 0987.81005 [13] Dvurečenskij, A, Perfect effect algebras are categorically equivalent with abelian interpolation po-groups, J. Aust. Math. Soc., 82, 183-207, (2007) · Zbl 1117.06009 [14] Figallo, A.V., Pelaitay, G.: Tense operators on De Morgan algebras. Logic J. IGPL (2013). doi:10.1093/jigpal/jzt024 · Zbl 1347.06012 [15] Foulis, DJ; Bennett, MK, Effect algebras and unsharp quantum logics, Found. Phys., 24, 1325-1346, (1994) · Zbl 1213.06004 [16] Paseka, J; Janda, J, A dynamic effect algebras with dual operation, Math. Appl., 1, 79-89, (2012) · Zbl 1296.03039 [17] Paseka, J, Operators on MV-algebras and their representations, Fuzzy Sets Syst., 232, 62-73, (2013) · Zbl 1314.06016 [18] PulmannovÁ, S., Pták, P.: Orthomodular Structures as Quantum Logics. Kluwer Academic Publishers, Dordrecht (1991) · Zbl 0743.03039 [19] Riečanová, Z, Generalization of blocks for D-lattices and lattice-ordered effect algebras, Int. J. Theor. Phys., 39, 231-237, (2000) · Zbl 0968.81003 [20] Riečanová, Z, Basic decomposition of elements and Jauch-piron effect algebras, Fuzzy Sets Syst., 155, 138-149, (2005) · Zbl 1073.81014 [21] Rutledge, J.D.: A preliminary investigation of the in nitely many-valued predicate calculus. Ph.D. thesis, Cornell University (1959) · Zbl 1296.03039 [22] Teheux, B, A duality for the algebras of a łukasiewicz $$n+1$$-valued modal system, Stud. Log., 87, 13-36, (2007) · Zbl 1127.03050 [23] Teheux B.: Algebraic approach to modal extensions of Łukasiewicz logics. Doctoral thesis, Université de Liege. http://orbi.ulg.ac.be/handle/2268/10887 (2009)
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