{"paper":{"title":"Semantical conditions for the definability of functions and relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Diego Vaggione, Miguel Campercholi","submitted_at":"2015-06-24T19:04:10Z","abstract_excerpt":"Let $\\mathcal{L}\\subseteq \\mathcal{L}^{\\prime }$ be first order languages, let $R\\in \\mathcal{L}^{\\prime }-\\mathcal{L}$ be a relation symbol, and let $% \\mathcal{K}$ be a class of $\\mathcal{L}^{\\prime }$-structures. In this paper we present semantical conditions equivalent to the existence of an $\\mathcal{L}$-formula $\\varphi \\left( \\vec{x}\\right) $ such that $\\mathcal{K}\\vDash \\varphi (\\vec{x})\\leftrightarrow R(\\vec{x})$, and $\\varphi $ has a specific syntactical form (e.g., quantifier free, positive and quantifier free, existential horn, etc.). For each of these definability results for rela"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.07501","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}