Uniform Birkhoff

We show that pseudovarieties of finitely generated algebras, i.e., classes C of finitely generated algebras closed under finite products, homomorphic images, and subalgebras, can be described via a uniform structure UC on the free algebra for C: the members of C then are precisely those finitely generated algebras A for which the natural mapping from the free algebra onto the term clone of A is well-defined and uniformly continuous with respect to the uniformity UC and the uniformity of pointwise convergence on the term clone of A, respectively. Our result unifies earlier theorems describing pseudovarieties of finite algebras and the pseudovariety generated by a single oligomorphic algebra.