Uniform Birkhoff

Michael Pinsker Mai Gehrke 1
1 LJAD
JAD - Laboratoire Jean Alexandre Dieudonné
Abstract : We show that pseudovarieties of finitely generated algebras, i.e., classes C of finitely generated algebras closed under finite products, homomorphic images, and subalge-bras, can be described via a uniform structure U C 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 U C 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, respectively.
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Michael Pinsker, Mai Gehrke. Uniform Birkhoff. Journal of Pure and Applied Algebra, Elsevier, 2018, 222 (5), pp.1242-1250. ⟨10.1016/j.jpaa.2017.06.016⟩. ⟨hal-02405510⟩

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