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The Logic of Lattice Effect Algebras Based on Induced Groupoids

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CHAJDA Ivan LAENGER Helmut PASEKA Jan

Rok publikování 2019
Druh Článek v odborném periodiku
Časopis / Zdroj JOURNAL OF MULTIPLE-VALUED LOGIC AND SOFT COMPUTING
Fakulta / Pracoviště MU

Přírodovědecká fakulta

Citace
www https://www.oldcitypublishing.com/journals/mvlsc-home/mvlsc-issue-contents/mvlsc-volume-33-number-3-2019/mvlsc-33-3-p-161-175/
Klíčová slova D-poset; effect algebra; lattice effect algebra; antitone involution; effect groupoid; groupoid-based logic
Popis Effect algebras were introduced by Foulis and Bennett as the so-called quantum structures which describe quantum effects and are determined by the behaviour of bounded self-adjoint operators on the phase space of the corresponding physical system which is a Hilbert space. From the algebraic point of view, the problem is that effect algebras are only partial ones and hence there can be drawbacks when we apply them for a construction of algebraic semantics of the corresponding logic of quantum mechanics. If the effect algebra in question is lattice-ordered this disadvantage can be overcome by using a representation of an equivalent algebra with everywhere defined operations. In our paper, this algebra is a groupoid equipped with one more unary operation which is an antitone involution. It enables us to introduce suitable axioms and inherence rules for the algebraic semantics of the corresponding logic and to prove that this logic is sound and complete.
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