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Compactness in equational logic

Weaver G.

Reports on Mathematical Logic

2005

Nr 39

25-46

Three versions of the compactness theorem for finitary equational logics are generalized and formulated as properties of infinitary equational logics and it is shown that these properties are coextensive. The central construction of the paper is derived from a model theoretic proof of the consequence version of the compactness theorem: if an equation is a consequence of a set of equations $S$, then it is a consequence of some finite subset of $S$. Modifying Birkhoff's 1935 construction, the terms $t$ and $t^{\prime}$ are related provided the equation $t = t^{\prime}$ is a consequence of some finite subset of $S$. This relation is a congruence on the algebra of terms and the quotient algebra induced by this congruence is a model of exactly those equations which are a consequence of some finite subset of $S$. This construction is extended to infinitary equational logics to show that if $\kappa$ is a regular cardinal strictly greater than the degree of each functional constant in the non-logical vocabulary of the language and no greater than the cardinality of the language, then any equation that is a consequence of $S$ is a consequence of a subset of $S$ of cardinality strictly less than $\kappa$.