Integrally closed domains with monomial presentations

Let A be a finitely generated commutative algebra over a field K with a presentation A = K〈X1, . . . ,Xn | R〉, where R is a set of monomial relations in the generators X1, . . . ,Xn. Necessary and sufficient conditions are found for A to be an integrally closed domain provided that the presentation involves at most two relations. The class group of such algebras A is calculated. Examples are given to indicate problems with an extension of these results to the case of more defining relations.