  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 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.
