The Robinson property and amalgamations of higher arities

Abstract

In this article we discuss a version of the Robinson property studied recently by Gyenis in , and we present a solution to one of his open problems. We say that a first‐order structure scriptM satisfies the Robinson property whenever the union of two non‐trivial partial n‐types over different finite sets is realizable if and only if they are not explicitly contradictory. In his article, Gyenis showed that a universal, homogeneous structure over a language that consists of at most binary relation symbols satisfies the Robinson property if and only if its age has a generalized amalgamation property (the so called prescribed amalgamation property, PAP). We go further and define a series of new kinds of amalgamation properties, APn for any natural number n. Using these properties we can characterize all universal and homogeneous structures that satisfy the Robinson property independently from the arities of relation symbols of the language.

The Modelwise Interpolation Property of Semantic Logics

“…Motivated by model theoretic investigations of homogeneous structures [15,27] the local interpolation property (lIP , for short) has been introduced in [16] as the property that whenever φ, ψ ∈ F for which ( ) φ,ψ holds, for all M ∈ M if M |= φ ψ, then there exists χ ∈ F with Voc(χ) ⊆ Voc(φ) ∩ Voc(ψ) such that M |= φ χ and M |= χ ψ. Notice that the lIP differs from the mIP in that in the former the implication φ ψ is also "localized" to models, making it a rather weak property of a logic.…”

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“…The aim of this paper is to give an algebraic characterization of a local version of Craig's interpolation theorem that has been earlier studied in [12], [17]. We use the framework of universal algebraic logic as introduced in [1,2].…”