{"paper":{"title":"Combinatorial characterization of pseudometrics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"J. Luukkainen, O. Dovgoshey","submitted_at":"2019-06-18T07:20:22Z","abstract_excerpt":"Let $X$, $Y$ be sets and let $\\Phi$, $\\Psi$ be mappings with the domains $X^{2}$ and $Y^{2}$ respectively. We say that $\\Phi$ is combinatorially similar to $\\Psi$ if there are bijections $f \\colon \\Phi(X^2) \\to \\Psi(Y^{2})$ and $g \\colon Y \\to X$ such that $\\Psi(x, y) = f(\\Phi(g(x), g(y)))$ for all $x$, $y \\in Y$. It is shown that the semigroups of binary relations generated by sets $\\{\\Phi^{-1}(a) \\colon a \\in \\Phi(X^{2})\\}$ and $\\{\\Psi^{-1}(b) \\colon b \\in \\Psi(Y^{2})\\}$ are isomorphic for combinatorially similar $\\Phi$ and $\\Psi$. The necessary and sufficient conditions under which a given "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.07411","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1906.07411/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}