Monoid algebras and graph products

TL;DR

This paper extends algebraic results about graph products, including Cartesian, strong, and direct products, to more complex graph constructions using monoid algebras and power series rings.

Contribution

It introduces new algebraic techniques to analyze rooted hierarchical and modified lexicographic graph products, broadening previous results to graphs with infinitely many components.

Findings

Unique nth roots and cancellation properties hold for extended graph products.

Results apply to graphs with countably many finite connected components under certain conditions.

Proofs utilize monoid algebras and generalized power series rings.

Abstract

In this note, we extend results about unique $n^{\textrm{th}}$ roots and cancellation of finite disconnected graphs with respect to the Cartesian, the strong and the direct product, to the rooted hierarchical products, and to a modified lexicographic product. We show that these results also hold for graphs with countably many finite connected components, as long as every connected component appears only finitely often (up to isomorphism). The proofs are via monoid algebras and generalized power series rings.