The logic induced by effect algebras
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The logic induced by effect algebras Ivan Chajda1 · Radomír Halaš1 · Helmut Länger1,2
© The Author(s) 2020
Abstract Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras E, we investigate a natural implication and prove that the implication reduct of E is term equivalent to E. Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras. Keywords Lattice effect algebra · Lattice effect implication algebra · Effect algebra · Effect implication algebra · Finite effect algebra · Gentzen system · Algebraic semantics Mathematics Subject Classification 03G12 · 03G25 · 06A11 · 06F99
1 Introduction Effect algebras were introduced by D. Foulis and M. K. Bennett (Foulis and Bennett 1994; see also Bennett and Foulis 1995 and Botur and Halaš 2009) as an algebraic axiomaCommunicated by A. Di Nola. Support of the research by ÖAD, Project CZ 02/2019, entitled “Function algebras and ordered structures related to logic and data fusion,” support of the research of the first and second author by IGA, Project PˇrF 2020 014, and support of the research of the first and third author by the Austrian Science Fund (FWF), Project I 4579-N, and the ˇ Czech Science Foundation (GACR), Project 20-09869L, entitled “The many facets of orthomodularity,” are gratefully acknowledged.
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Helmut Länger [email protected] Ivan Chajda [email protected] Radomír Halaš [email protected]
1
Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic
2
Institute of Discrete Mathematics and Geometry, Faculty of Mathematics and Geoinformation, TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria
tization of the logic of quantum mechanics. For enabling deductions and derivations in this logic, it is necessary to introduce the connective implication. It is worth noticing that for lattice effect algebras this task was solved in a different way in Chajda et al. (2017). The problem is that though the binary operation of an effect algebra is only partial, implication should be defined everywhere. If the considered effect algebra is lattice-ordered, then implication is usually defined by x → y := x + (x ∧ y) and called Sasaki implication, see e.g., Borzooei et al. (2018), Chajda and Länger (2019), Foulis and Pulmannová (2012) and Rad et al. (2019). Properties of this implication were described in these papers, and a certain
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