somorphism problems for finite dimensional algebras can be computationally hard. When the algebras are monomial, it is shown, refining work of Shirayanagi, that there is a simple definitive combinatorial method. However, examples show that no such criterion is possible if the class of algebras is expanded to that of diagram algebras (in the sense of Fuller). The presentation of a diagram algebra is field independent but the existence of an isomorphism between two such is not. (Subject classes: 16G30, 16P10, 20M25).