Uniform Birkhoff

Mai Gehrke (1) , Michael Pinsker (2)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 TU Wien - Institute of Discrete Mathematics and Geometry [Vienne]

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.

Domaines: Mathématiques [math]; Anneaux et algèbres [math.RA]; Informatique [cs]

Mai Gehrke, Michael Pinsker. Uniform Birkhoff. Journal of Pure and Applied Algebra, 2018, 222 (5), pp.1242-1250. ⟨10.1016/j.jpaa.2017.06.016⟩. ⟨hal-02405510⟩