A Generalized Vandermonde Determinant

We prove two determinantal identities that generalize the Vandermondedeterminant identity . In the first of our identities the set {0, ..., m} indexing the rows and columns of thedeterminant is replaced by an arbitrary finite order ideal in the set ofsequences of nonnegative integers which are 0 except for a finite numberof components. In the second the index set is replaced by an arbitraryfinite order ideal in the set of all partitions.