Simplicial effects and weakly associative partial groups

TL;DR

This paper introduces a new category of simplicial effects extending effect algebras by relaxing associativity, capturing structures relevant to distributions and measurements in quantum theory.

Contribution

It develops a framework for simplicial effects and weakly associative partial groups as extensions of effect algebras, broadening the algebraic structures used in quantum measurement theory.

Findings

Defines simplicial effects as an extension of effect algebras.

Characterizes weakly associative partial groups within the framework.

Connects the structures to simplicial distributions and measurements.

Abstract

In this paper, we introduce a new category of simplicial effects that extends the categories of effect algebras and their multi-object counterpart, effect algebroids. Our approach is based on relaxing the associativity condition satisfied by effect algebras and, more generally, partial monoids. Within this framework, simplicial effects and weakly associative partial groups arise as two extreme cases in the category of weak partial monoids. Our motivation is to capture simplicial structures from the theory of simplicial distributions and measurements that behave like effects.