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Motivated by model-theoretic properties of the BSR class, we present a familyof semantic classes of FO formulae with finite or co-finite spectra over arelational vocabulary \Sigma. A class in this family is denotedEBS_\Sigma(\sigma), where \sigma is a subset of \Sigma. Formulae inEBS_\Sigma(\sigma) are preserved under substructures modulo a bounded core andmodulo re-interpretation of predicates outside \sigma. We study properties ofthe family EBS_\Sigma = {EBS_\Sigma(\sigma) | \sigma \subseteq \Sigma}, e.g.classes in EBS_\Sigma are spectrally indistinguishable, EBS_\Sigma(\Sigma) issemantically equivalent to BSR over \Sigma, and EBS_\Sigma(\emptyset) is theset of all FO formulae over \Sigma with finite or co-finite spectra.Furthermore, (EBS_\Sigma, \subseteq) forms a lattice isomorphic to the powersetlattice (\wp({\Sigma}), \subseteq). This gives a natural semanticgeneralization of BSR as ascending chains in (EBS_\Sigma, \subseteq). Manywell-known FO classes are semantically subsumed by EBS_\Sigma(\Sigma) orEBS_\Sigma(\emptyset). Our study provides alternative proofs of interestingresults like the Lo\'s-Tarski Theorem and the semantic subsumption of theL\"owenheim class with equality by BSR. We also present a syntactic sub-classof EBS_\Sigma(\sigma) called EDP_\Sigma(\sigma) and give an expression for thesize of the bounded cores of models of EDP_\Sigma(\sigma) formulae. We showthat the EDP_\Sigma(\sigma) classes also form a lattice structure. Finally, westudy some closure properties and applications of the classes presented.

On Semantic Generalizations of the Bernays-Sch\"onfinkel-Ramsey Class with Finite or Co-finite Spectra