Sum of Observables on MV-Effect Algebras

TL;DR

This paper introduces a new way to sum bounded observables in $\sigma$-MV-effect algebras, revealing algebraic structures like semigroups and $ ext{l}$-groups, enhancing the mathematical framework of quantum observables.

Contribution

It defines the sum of bounded observables via spectral resolutions and explores the algebraic and order structures they form in $\sigma$-MV-effect algebras.

Findings

Sum of two bounded observables is commutative and associative.

Set of bounded observables forms a partially ordered semigroup.

Sharp observables form a Dedekind $ ext{σ}$-complete $ ext{l}$-group.

Abstract

Using a one-to-one correspondence between observables and their spectral resolutions, we introduce the sum of any two bounded observables of a $\sigma$-MV-effect algebra. This sum is commutative, associative and with neutral element. Under the Olson order of observables, the set of bounded observables is a partially ordered semigroup, and the set of sharp observables is even a Dedekind $\sigma$-complete $\ell$-group with strong unit.