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.

中文翻译：

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由效应代数引起的逻辑。

效应代数形成了量子力学逻辑的代数形式。对于格效应代数E，我们研究了一个自然蕴涵，并证明E的蕴涵约简等于E。然后，我们提出一种简单的Gentzen式公理系统，以公理化由格效应代数引起的逻辑。对于不需要代数排序的效果代数，我们引入某种隐式，该隐式随处可见，但其结果不必是单个元素。然后，我们研究了效果蕴涵代数，并证明了这些代数与满足上升链条件的效果代数之间的对应关系。