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algebra universalis

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

The Birkhoff theorem for finite algebras

  • Jan Reiterman
Article

Abstract

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.

Keywords

Finite Type Homomorphic Image Unary Operation Algebra UNIV Free Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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