Homomorphisms and chains of Kripke models
This paper is a continuation of the study of preservation theorems in intuitionistic logic by the authors [``Preservation theorems for Kripke models'', Math. Log. Q. 55, No. 2, 177--184 (2009; Zbl 1160.03018)]. It is based on Kripke semantics and restricts to those Kripke models defined on a fixed frame \(F\). A homomorphism (resp. embedding) between two Kripke models is a compatible family of maps between the corresponding nodes which preserves atomic formulas from left to right (resp. from both sides). Two classes of formulas \(\mathcal U(\Phi,\Psi)\) and \(\mathcal E(\Phi,\Psi)\) are introduced which extend the ordinary classes of universal and existential formulas. Theories axiomatized by sentences from each one are characterized in terms of homomorphisms and embeddings, similar to the classical cases. Also, a class of theories preserved under the union of chain is introduced. Finally, as an application to intuitionistic arithmetic, an independence result of Wehmeier is strengthened.
scientific article; zbMATH DE number 5886781