# Properties of the connective implication in effect algebras

## TL;DR

This paper introduces and analyzes a connective implication within effect algebras, linking it to conjunction through unsharp residuation, and explores its logical properties and rules in the context of quantum logic.

## Contribution

It defines a new implication connective in effect algebras, studies its properties, and connects it with unsharp residuation, advancing the algebraic semantics of quantum logic.

## Key findings

- The implication is connected with conjunction via unsharp residuation.
- The structure can be converted back into an effect algebra, ensuring soundness.
- The study of Modus Ponens and contraposition law in this framework.

## Abstract

Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serves as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is is equipped with a connective implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain "unsharp" residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.