Amalgamation, Interpolation and Epimorphisms in Algebraic Logic

Abstract

The amalgamation property (for classes of models), since its discovery, has played a dominant role in algebra and model theory. Algebraic logic is the natural interface between universal algebra and logic (in our present context a variant of first order logic). Indeed, in algebraic logic amalgamation properties in classes of algebras are proved to be equivalent to interpolation results in the corresponding logic. In algebra, the properties of epimorphisms (in the categorial sense) being surjective is well studied. In algebraic logic, this property is strongly related to the famous Beth definability property [Hen-Mon-Tar,85] 5.6.10. Pigozzi [Pig,71] is a milestone for working out such equivalences for cylindric algebras, see also [Mad-Say,07], [And-Nem-Sai,01]. In first order logic, Craig interpolation theorem is equivalent to Beth definability theorem; however this equivalence no longer holds for certain modifications of first order logic, be it reducts or expansions, like L _n (first order logic restricted to finitely many n variables) and the extensions of first order logic studied in [Hen-Mon-Tar,85] Sec. 4.3, which are essentially finitary but have an infinitary flavour.

The work of Madarász was supported by a Bolyai grant and by OTKA grant T81188.