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M. Pinsker, M. Gehrke:
"Uniform Birkhoff";
Journal of Pure and Applied Algebra, 222 (2018), 5; S. 1242 - 1250.

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.

http://dx.doi.org/10.1016/j.jpaa.2017.06.016