Abstract
Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras {mathbf {E}}, we investigate a natural implication and prove that the implication reduct of {mathbf {E}} is term equivalent to {mathbf {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.
Highlights
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 PrF 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 Czech Science Foundation (GAC R), Project 20-09869L, entitled “The many facets of orthomodularity,” are gratefully acknowledged
We prove that lattice effect algebras and lattice effect implication algebras are term equivalent and in a natural one-to-one correspondence
Let us mention that a system of axioms and rules for the propositional logic induced by lattice effect algebras was already presented in Rad et al (2019)
Summary
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 PrF 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 Czech Science Foundation (GAC R), Project 20-09869L, entitled “The many facets of orthomodularity,” are gratefully acknowledged Another reason for introducing implication in effect algebras is to show that this connective is related to conjunction via (left) adjointness, and effect algebras can be considered as left residuated structures; see Chajda and Länger (2019), Chajda and Länger (submitted). In the second part of the paper, we extend our investigations to not necessarily lattice-ordered effect algebras satisfying the ascending chain condition, in particular to finite effect algebras In this case, we characterize the operation of implication in a similar way as it was done in the lattice case.
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