On Semantic Generalizations of the Bernays-Schönfinkel-Ramsey Class with Finite or Co-finite Spectra

Motivated by model-theoretic properties of the BernaysSchönfinkel-Ramsey (BSR) class, we present a family of semantic classes of FO formulae with finite or co-finite spectra over a relational vocabulary Σ. A class in this family is denoted EBSΣ(σ), where σ ⊆ Σ. Formulae in EBSΣ(σ) are preserved under substructures modulo a bounded core and modulo re-interpretation of predicates in Σ \ σ. We study several properties of the family EBSΣ = {EBSΣ(σ) | σ ⊆ Σ}. For example, classes in EBSΣ are spectrally indistinguishable, the smallest class, EBSΣ(Σ), is semantically equivalent to BSR over Σ, and the largest class, EBSΣ(∅), is the set of all FO formulae over Σ with finite or co-finite spectra. Furthermore, (EBSΣ,⊆) forms a lattice that is isomorphic to the powerset lattice (℘(Σ),⊆). We also show that if Σ contains at least one predicate of arity ≥ 2, there exist semantic gaps between EBSΣ(σ1) and EBSΣ(σ2) if σ1 6= σ2. This gives a natural semantic generalization of BSR as ascending chains in the lattice