  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.
