Simplicial Effects and Weakly Associative Partial Groups

This paper introduces a broad generalization of effect algebras and of their multi-object counterparts effect algebroids, with the ultimate aim of providing a unified mathematical structure capable of capturing the simplicial, effect-algebra-like structures which arise from the theory of simplicial distributions. The key innovation of this paper is a weakened version of the associativity conditions satisfied by effect algebras and partial monoids, which allows us fit a much broader range of examples—in particular, many interesting classes of partial groups—into our framework. Indeed, within the rubric of weak associativity, simplicial effects and weakly associative partial groups arise as two extreme cases in the category of weak partial monoids. We conclude the paper by providing an example of a simplicial effect which is not an effect algebroid and showing that Gleason’s Theorem still holds for this simplicial effect.