{"paper":{"title":"Definability in the embeddability ordering of finite directed graphs, II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"\\'Ad\\'am Kunos","submitted_at":"2018-06-20T17:57:40Z","abstract_excerpt":"We deal with first-order definability in the embeddability ordering $( \\mathcal{D}; \\leq)$ of finite directed graphs. A directed graph $G\\in \\mathcal{D}$ is said to be embeddable into $G' \\in \\mathcal{D}$ if there exists an injective graph homomorphism $\\varphi \\colon G \\to G'$. We describe the first-order definable relations of $( \\mathcal{D}; \\leq)$ using the first-order language of an enriched small category of digraphs. The description yields the main result of one of the author's papers as a corollary and a lot more. For example, the set of weakly connected digraphs turns out to be first-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.07871","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"}