Chajda Ivan, Halaš Radomír, Länger Helmut
Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic.
Institute of Discrete Mathematics and Geometry, Faculty of Mathematics and Geoinformation, TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria.
Soft comput. 2020;24(19):14275-14286. doi: 10.1007/s00500-020-05188-w. Epub 2020 Jul 26.
Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras , we investigate a natural implication and prove that the implication reduct of is term equivalent to . 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.
效应代数构成了量子力学逻辑的一种代数形式化。对于格效应代数，我们研究一种自然蕴含关系，并证明其蕴含归约与[具体内容未给出]是项等价的。然后，我们给出一个简单的根岑风格公理系统，以便对由格效应代数诱导的逻辑进行公理化。对于不一定是格序的效应代数，我们引入一种处处有定义但其结果不一定是单个元素的蕴含关系。然后，我们研究效应蕴含代数，并证明这些代数与满足升链条件的效应代数之间的对应关系。我们也为不一定是格序的效应代数给出一个根岑风格的公理系统，并证明它是由有限效应代数诱导的逻辑的一种代数语义。
