The groupoid-based logic for lattice effect algebras

TL;DR

This paper introduces a new logical framework for lattice effect algebras by establishing a correspondence with certain groupoids, enabling a groupoid-based representation and logic for these algebraic structures.

Contribution

It provides the first groupoid representation of lattice effect algebras and develops a corresponding logic based on this structure.

Findings

Established a one-to-one correspondence between lattice effect algebras and groupoids with involution.

Developed a new logic for lattice effect algebras using the groupoid representation.

Provided a foundational link between algebraic structures and logical systems.

Abstract

The aim of the paper is to establish a certain logic corresponding to lattice effect algebras. First, we answer a natural question whether a lattice effect algebra can be represented by means of a groupoid-like structure. We establish a one-to-one correspondence between lattice effect algebras and certain groupoids with an antitone involution. Using these groupoids, we are able to introduce a suitable logic for lattice effect algebras.