algebra universalis

, Volume 14, Issue 1, pp 1–10 | Cite as

The Birkhoff theorem for finite algebras

  • Jan Reiterman


A finite analogue of the Birkhoff variety theorem is proved: a non-void class of finite algebras of a finite type τ is closed under the formation of finite products, subalgebras and homomorphic images if and only if it is definable by equations for implicit operations, that is, roughly speaking, operations which are not necessarily induced by τ-terms but which are compatible with all homomorphisms. It is well-known that explicit operations (those induced by τ-terms) do not suffice for such an equational description. Topological aspects of implicit operations are considered. Various examples are given.


Finite Type Homomorphic Image Unary Operation Algebra UNIV Free Algebra 
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Copyright information

© Birkhäuser Verlag 1982

Authors and Affiliations

  • Jan Reiterman
    • 1
  1. 1.Technical University of PraguePragueCzechoslovakia

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