{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:XXTZMSANFXZWVOMNA55NHPTUGF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"fff502146555b1289e733652f13019e68e14edcd262cb696e0de911d4b802bd5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-05T15:44:35Z","title_canon_sha256":"65bf2988fa7982f1d1da091aaf339dc08a7b0bfcdbee661d0b545b71a3486326"},"schema_version":"1.0","source":{"id":"1003.1294","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1003.1294","created_at":"2026-05-18T00:57:38Z"},{"alias_kind":"arxiv_version","alias_value":"1003.1294v1","created_at":"2026-05-18T00:57:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1003.1294","created_at":"2026-05-18T00:57:38Z"},{"alias_kind":"pith_short_12","alias_value":"XXTZMSANFXZW","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_16","alias_value":"XXTZMSANFXZWVOMN","created_at":"2026-05-18T12:26:17Z"},{"alias_kind":"pith_short_8","alias_value":"XXTZMSAN","created_at":"2026-05-18T12:26:17Z"}],"graph_snapshots":[{"event_id":"sha256:99955a439643626f12ca913297dccaeb4778494ddea8775dad3ba3d2a0b648ac","target":"graph","created_at":"2026-05-18T00:57:38Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the arity gap of functions of several variables defined on an arbitrary set A and valued in another set B. The arity gap of such a function is the minimum decrease in the number of essential variables when variables are identified. We establish a complete classification of functions according to their arity gap, extending existing results for finite functions. This classification is refined when the codomain B has a group structure, by providing unique decompositions into sums of functions of a prescribed form. As an application of the unique decompositions, in the case of finite sets","authors_text":"Erkko Lehtonen, Miguel Couceiro, Tam\\'as Waldhauser","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-05T15:44:35Z","title":"Decompositions of functions based on arity gap"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.1294","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3abb7dd568f2aa666d4ad62613a6eda598b2ae9e8fbdc486b732462f741cfb6a","target":"record","created_at":"2026-05-18T00:57:38Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"fff502146555b1289e733652f13019e68e14edcd262cb696e0de911d4b802bd5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-03-05T15:44:35Z","title_canon_sha256":"65bf2988fa7982f1d1da091aaf339dc08a7b0bfcdbee661d0b545b71a3486326"},"schema_version":"1.0","source":{"id":"1003.1294","kind":"arxiv","version":1}},"canonical_sha256":"bde796480d2df36ab98d077ad3be743147d97dd143273fd8ec19b049f91fbb1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bde796480d2df36ab98d077ad3be743147d97dd143273fd8ec19b049f91fbb1a","first_computed_at":"2026-05-18T00:57:38.744481Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:57:38.744481Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+ypTqh7Z3lXRdX/At4HdwHZYaEDUUaUFrHprL0+x/daPtZxgJZLeAWGDw68dpzkpsavcqHILowCK0BNpkUZYDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:57:38.745144Z","signed_message":"canonical_sha256_bytes"},"source_id":"1003.1294","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3abb7dd568f2aa666d4ad62613a6eda598b2ae9e8fbdc486b732462f741cfb6a","sha256:99955a439643626f12ca913297dccaeb4778494ddea8775dad3ba3d2a0b648ac"],"state_sha256":"5345fc7f52fdd320e71dd2c3cd9add46b38fb09093e490c89e7f9a18993fca5c"}