{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:R2WOUROUVKXZPQELW5PCT76VPI","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":"c3d4b7be0b7127bfda97e5c42862f9988de5e9891b23be261aef757b3b1d5c1b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-03-11T13:11:17Z","title_canon_sha256":"647c0d1c92714d9af13f2bb864dd94197d3a0f70d60b45d38a7d9966d3d18fb8"},"schema_version":"1.0","source":{"id":"1403.2564","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.2564","created_at":"2026-05-18T01:12:18Z"},{"alias_kind":"arxiv_version","alias_value":"1403.2564v1","created_at":"2026-05-18T01:12:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.2564","created_at":"2026-05-18T01:12:18Z"},{"alias_kind":"pith_short_12","alias_value":"R2WOUROUVKXZ","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_16","alias_value":"R2WOUROUVKXZPQEL","created_at":"2026-05-18T12:28:46Z"},{"alias_kind":"pith_short_8","alias_value":"R2WOUROU","created_at":"2026-05-18T12:28:46Z"}],"graph_snapshots":[{"event_id":"sha256:0833224834d730777d65812aefd034692a2fe855a7dc096cd5e1f9aaa34ffb4e","target":"graph","created_at":"2026-05-18T01:12:18Z","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":"A groupoid G is called an AG-groupoid if it satisfies the left invertive law: (ab)c = (cb)a. An AG-group G, is an AG-groupoid with left identity e \\in G (that is, ea = a for all a \\in G) and for all a \\in G there exists a' \\in G such that a.a' = a'.a = e. In this article we introduce the concept of AG-groupoids (mod n) and AG-group (mod n) using Vasantha's constructions [1]. This enables us to prove that AG-groupoids (mod n) and AG-groups (mod n) exist for every integer n \\geq 3. We also give some nice characterizations of some classes of AG-groupoids in terms of AG-groupoids (mod n).","authors_text":"Amanullah, Imtiaz Ahmad, Muhammad Rashad, Muhammad Shah","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-03-11T13:11:17Z","title":"On Modulo AG-groupoids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.2564","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:fbd30755b17f3f34d90ac18808b3a5dd5f2c09868f59b4bc9a78a261b276d139","target":"record","created_at":"2026-05-18T01:12:18Z","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":"c3d4b7be0b7127bfda97e5c42862f9988de5e9891b23be261aef757b3b1d5c1b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-03-11T13:11:17Z","title_canon_sha256":"647c0d1c92714d9af13f2bb864dd94197d3a0f70d60b45d38a7d9966d3d18fb8"},"schema_version":"1.0","source":{"id":"1403.2564","kind":"arxiv","version":1}},"canonical_sha256":"8eacea45d4aaaf97c08bb75e29ffd57a013bb32ff0f904781b29642458ced3bb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8eacea45d4aaaf97c08bb75e29ffd57a013bb32ff0f904781b29642458ced3bb","first_computed_at":"2026-05-18T01:12:18.815692Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:18.815692Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6pAHD0KpQFGfpkmYYiBw2l2xetDbbENkIb6IcZba7zAodcFLgxDubwsbdEMQtT8VriACP5KDKJtb8ohMCzFIAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:18.816046Z","signed_message":"canonical_sha256_bytes"},"source_id":"1403.2564","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fbd30755b17f3f34d90ac18808b3a5dd5f2c09868f59b4bc9a78a261b276d139","sha256:0833224834d730777d65812aefd034692a2fe855a7dc096cd5e1f9aaa34ffb4e"],"state_sha256":"c2e741c5a53b50471fad276394a97ba4e33eaf316972f83aa1a4b43832f7ae58"}