Preservation and Axiomatizability
Axiomatizability results express a rather subtle relationship between semantics and syntax. They give complete characterizations of certain classes of theories in purely semantic terms, formulated as closure properties of classes of models under some categorical operators. Perhaps the most famous example is the Birkhoff Variety theorem of equational logic: a class of algebras for a signature is closed under products, sub-algebras, and homomorphic images if and only if it is the class of algebras of an equational theory.
KeywordsPartial Algebra Inclusion System Elementary Extension Elementary Diagram Existential Sentence
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