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Then there exists a family $\\mathcal{F}$ of bijective selectors of $F$ such that $|\\mathcal{F}|<\\kappa$ and $F(x) = \\{ f(x): f\\in\\mathcal{F}\\}$ for each $x\\in X$. We apply this result to $G$-space representations of balleans."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1908.03911","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2019-08-11T14:11:22Z","cross_cats_sorted":[],"title_canon_sha256":"26f3c987d62e54297c970c445ec683b5e9d176924c68dd829c01d2dcc0adb5b6","abstract_canon_sha256":"904a53cd0cbb5c4c643690f3e6db6c4c48c78e380c44589766ccbd3f9f8a19de"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:15:49.809605Z","signature_b64":"u/bwgwk7Iuy3GoiM84cBHlv1k6ZV5q5XIDND27+xKcLZFgsE2pbLvnxUtFvJWv3pOXKdEmdzu3OUgIVXD7sYBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15209da668e338e3058b5d7dd8a951ef6be2be9e8bbdf251cfe9d338cafa6555","last_reissued_at":"2026-07-05T00:15:49.809158Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:15:49.809158Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decompositions of set-valued mappings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"Igor Protasov","submitted_at":"2019-08-11T14:11:22Z","abstract_excerpt":"Let $X$ be a set, $B_{X}$ denotes the family of all subsets of $X$ and $F: X \\longrightarrow B_{X}$ be a set-valued mapping such that $x \\in F(x)$, $sup_{x\\in X} | F(x)|< \\kappa$, $sup_{x\\in X} | F^{-1}(x)|< \\kappa$ for all $x\\in X$ and some infinite cardinal $\\kappa$. Then there exists a family $\\mathcal{F}$ of bijective selectors of $F$ such that $|\\mathcal{F}|<\\kappa$ and $F(x) = \\{ f(x): f\\in\\mathcal{F}\\}$ for each $x\\in X$. We apply this result to $G$-space representations of balleans."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.03911","kind":"arxiv","version":2},"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/1908.03911/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.03911","created_at":"2026-07-05T00:15:49.809221+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.03911v2","created_at":"2026-07-05T00:15:49.809221+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.03911","created_at":"2026-07-05T00:15:49.809221+00:00"},{"alias_kind":"pith_short_12","alias_value":"CUQJ3JTI4M4O","created_at":"2026-07-05T00:15:49.809221+00:00"},{"alias_kind":"pith_short_16","alias_value":"CUQJ3JTI4M4OGBML","created_at":"2026-07-05T00:15:49.809221+00:00"},{"alias_kind":"pith_short_8","alias_value":"CUQJ3JTI","created_at":"2026-07-05T00:15:49.809221+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CUQJ3JTI4M4OGBMLLV65RKKR55","json":"https://pith.science/pith/CUQJ3JTI4M4OGBMLLV65RKKR55.json","graph_json":"https://pith.science/api/pith-number/CUQJ3JTI4M4OGBMLLV65RKKR55/graph.json","events_json":"https://pith.science/api/pith-number/CUQJ3JTI4M4OGBMLLV65RKKR55/events.json","paper":"https://pith.science/paper/CUQJ3JTI"},"agent_actions":{"view_html":"https://pith.science/pith/CUQJ3JTI4M4OGBMLLV65RKKR55","download_json":"https://pith.science/pith/CUQJ3JTI4M4OGBMLLV65RKKR55.json","view_paper":"https://pith.science/paper/CUQJ3JTI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.03911&json=true","fetch_graph":"https://pith.science/api/pith-number/CUQJ3JTI4M4OGBMLLV65RKKR55/graph.json","fetch_events":"https://pith.science/api/pith-number/CUQJ3JTI4M4OGBMLLV65RKKR55/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CUQJ3JTI4M4OGBMLLV65RKKR55/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CUQJ3JTI4M4OGBMLLV65RKKR55/action/storage_attestation","attest_author":"https://pith.science/pith/CUQJ3JTI4M4OGBMLLV65RKKR55/action/author_attestation","sign_citation":"https://pith.science/pith/CUQJ3JTI4M4OGBMLLV65RKKR55/action/citation_signature","submit_replication":"https://pith.science/pith/CUQJ3JTI4M4OGBMLLV65RKKR55/action/replication_record"}},"created_at":"2026-07-05T00:15:49.809221+00:00","updated_at":"2026-07-05T00:15:49.809221+00:00"}