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.
KeywordsFinite Type Homomorphic Image Unary Operation Algebra UNIV Free Algebra
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