On graph products of monoids

TL;DR

This paper investigates algebraic properties of graph products of monoids, showing that regularity is not preserved but properties like abundance and Fountainicity are, and establishes a unique normal form for elements.

Contribution

It introduces the concept that graph products of regular monoids are abundant and extends closure results for left abundant and Fountain monoids, with a new normal form for elements.

Findings

Graph products of regular monoids are abundant.

Closure of left abundant and Fountain monoids under graph product.

Existence of a unique Foata normal form for elements in graph products.

Abstract

Graph products of monoids provide a common framework for direct and free products, and graph monoids (also known as free partially commutative monoids). If the monoids in question are groups, then any graph product is, of course, a group. For monoids that are not groups, regularity is perhaps the first and most important algebraic property that one considers; however, graph products of regular monoids are not in general regular. We show that a graph product of regular monoids satisfies the related, but weaker, condition of being abundant. More generally, we show that the classes of left abundant and left Fountain monoids are closed under graph product. The notions of abundancy and Fountainicity and their one-sided versions arise from many sources, for example, that of abundancy from projectivity of monogenic acts, and that of Fountainicity (also known as weak abundancy) from connections with ordered categories. As a very special case we obtain the earlier result of Fountain and Kambites that the graph product of right cancellative monoids is right cancellative. To achieve our aims we show that elements in (arbitrary) graph products have a unique Foata normal form, and give some useful reduction results; these may equally well be applied to groups as to the broader case of monoids.