Unsharp residuation in effect algebras

TL;DR

This paper introduces unsharp residuated posets, a new algebraic structure that generalizes effect algebras by allowing residuation with everywhere defined terms, even when the algebra is not lattice-ordered.

Contribution

It proposes the concept of unsharp residuated posets, extending effect algebras to include residuation with total terms and without requiring lattice order, enriching the algebraic framework for quantum logic.

Findings

Unsharp residuated posets can be organized into effect or pseudoeffect algebras.

The new structure accommodates non-lattice-ordered effect algebras.

It generalizes residuation concepts to partial and non-lattice contexts.

Abstract

Effect algebras and pseudoeffect algebras were introduced by Foulis, Bennett, Dvurecenskij and Vetterlein as so-called quantum structures which serve as an algebraic axiomatization of the logic of quantum mechanics. A natural question concerns their connections to substructural logics which are described by means of residuated lattices or posets. In a previous paper it was shown that an effect algebra can be organized into a so-called conditionally residuated structure where the adjointness condition holds only for those elements for which multiplication and implication are defined. Because this is a very strong restriction, we try to find another kind of residuation where the terms occurring in the adjointness condition are everywhere defined though the binary operation of a given effect algebra is only partial. Moreover, we work with effect algebras which need not be lattice-ordered and hence the lattice operations join and meet are replaced by means of upper and lower cones which, however, are not elements but subsets. Hence, the resulting concept, the so-called unsharp residuated poset, is equipped with LU-terms which substitute operations, but are everywhere defined. Although this concept seems rather complicated at the first glance, we prove that such an unsharp residuated poset can be conversely organized into an effect algebra or a pseudoeffect algebra depending on commutativity of the multiplication.