Heaps of Fish: arrays, generalized associativity and heapoids

TL;DR

This paper explores a ternary generalization of associativity using hypergraph calculus, introducing new structures like ternary categories and heapoids, and framing associativity as a hypergraph rewrite confluence.

Contribution

It introduces a diagrammatic hypergraph framework for ternary associativity, defining ternary categories and heapoids, and extends the theory of associativity to higher arity structures.

Findings

Defined a hypergraph calculus for ternary associativity

Introduced ternary categories and heapoids

Revealed associativity as a hypergraph rewrite confluence

Abstract

In this paper we investigate a ternary generalization of associativity by defining a diagrammatic calculus of hypergraphs that extends the usual notions of tensor networks, categories and relational algebras. In doing so we rediscover the ternary structures known as heaps and are able to give a more comprehensive treatment of their mergence in the context of dagger categories and their generalizations. Our key insight is to approach associativity as a confluence property of hypergraph rewrite systems. This approach allows us to define a notion of ternary category and heapoid, where morphisms bind three objects simultaneously, and suggests a systematic study of higher arity forms of associativity.