Lattice pseudo-effect algebras as double residuated structures

TL;DR

This paper explores lattice pseudo-effect algebras as non-commutative structures, characterizing them through double residuated structures and pseudo Sasaki algebras, advancing the algebraic understanding of non-commutative reasoning.

Contribution

It introduces a novel characterization of lattice pseudo-effect algebras via pseudo Sasaki algebras and double residuated structures, expanding the algebraic framework for non-commutative logic.

Findings

Lattice pseudo-effect algebras are characterized by pseudo Sasaki algebras.

All pseudo-effect algebras can be described using partially defined double residuated structures.

The study reveals doubled connectives in non-commutative reasoning, including two negations and two pairs of conjunction and implication.

Abstract

Pseudo-effect algebras are partial algebraic structures, that were introduced as a non-commutative generalization of effect algebras. In the present paper, lattice ordered pseudo-effect algebras are considered as possible algebraic non-commutative analogs of non-commutative non-standard reasoning. To this aim, the interplay among conjunction, implication and negation connectives is studied. It turns out that in the non-commutative reasoning, all these connectives are doubled. In particular, there are two negations and two pairs consisting of conjunction and implication, related by residuation laws. The main result of the paper is a characterization of lattice pseudo-effect algebras in terms of so-called pseudo Sasaki algebras. We also show that all pseudo-effect algebras can be characterized in terms of certain partially defined double residuated structures.