Associativity and non-associativity of some hypergraph products

TL;DR

This paper investigates various hypergraph products, identifying which are associative and providing counterexamples for non-associative variants, thus clarifying their algebraic properties.

Contribution

It determines the associativity of several hypergraph products, highlighting that only some variants are associative, including the Cartesian, minimal rank preserving direct, and normal products.

Findings

Only some hypergraph products are associative.

Counterexamples show non-associativity in others.

Clarifies algebraic properties of hypergraph products.

Abstract

Several variants of hypergraph products have been introduced as generalizations of the strong and direct products of graphs. Here we show that only some of them are associative. In addition to the Cartesian product, these are the minimal rank preserving direct product, and the normal product. Counter-examples are given for the strong product as well as the non-rank-preserving and the maximal rank preserving direct product.