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We define the disk with center $b$ and radius $a$, $\\mathcal{D}(a;b)$, as \\[\\mathcal{D}(a;b)=\\{x\\in \\Phi(r)+c\\mid r\\neq 0,\\ b\\in \\Phi(r)+c,\\ |(\\Phi(r)+c)\\cap (\\Phi(a)+b)|=1\\}.\\] We prove that in the field-generated case there are many analogies with the Euclidean geometry. Moreover, if $\\mathcal{B}^{\\mathcal{D}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure $(N,\\mathcal{B}^{\\mathcal{D}},\\in)$ is actually a balanced incomplete block design."},"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":"1012.1059","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-12-06T00:44:28Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"adabfdd72a91064945bd517e5a73f21e6cf711e404cbb174c966e43a33fc2e4c","abstract_canon_sha256":"418bcad166b0fb8c8d111721e89cf87ce98a4793e03e5a08657c1d71daf94382"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:12.016376Z","signature_b64":"I5rIxeIgrzxG5FGYENBGXV7f4NoMiYhYVFGPXa6CMhhWGxG2LyDIpFfxEKyuQnF41CYLucETkUD3x1/F5kSqDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4f27fce6bb030d24eb58fb43d9322b76d6a7a08ffbb01dba5cad0413222e0b7","last_reissued_at":"2026-05-18T04:02:12.015745Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:12.015745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Circular planar nearrings: geometrical and combinatorial aspects","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RA","authors_text":"Achille Frigeri, Anna Benini, Fiorenza Morini","submitted_at":"2010-12-06T00:44:28Z","abstract_excerpt":"Let $(N,\\Phi)$ be a circular Ferrero pair. We define the disk with center $b$ and radius $a$, $\\mathcal{D}(a;b)$, as \\[\\mathcal{D}(a;b)=\\{x\\in \\Phi(r)+c\\mid r\\neq 0,\\ b\\in \\Phi(r)+c,\\ |(\\Phi(r)+c)\\cap (\\Phi(a)+b)|=1\\}.\\] We prove that in the field-generated case there are many analogies with the Euclidean geometry. Moreover, if $\\mathcal{B}^{\\mathcal{D}}$ is the set of all disks, then, in some interesting cases, we show that the incidence structure $(N,\\mathcal{B}^{\\mathcal{D}},\\in)$ is actually a balanced incomplete block design."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.1059","kind":"arxiv","version":5},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1012.1059","created_at":"2026-05-18T04:02:12.015852+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.1059v5","created_at":"2026-05-18T04:02:12.015852+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.1059","created_at":"2026-05-18T04:02:12.015852+00:00"},{"alias_kind":"pith_short_12","alias_value":"YTZH7TTLWAYN","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"YTZH7TTLWAYNETVV","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"YTZH7TTL","created_at":"2026-05-18T12:26:17.028572+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/YTZH7TTLWAYNETVVR62D3EZCW5","json":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5.json","graph_json":"https://pith.science/api/pith-number/YTZH7TTLWAYNETVVR62D3EZCW5/graph.json","events_json":"https://pith.science/api/pith-number/YTZH7TTLWAYNETVVR62D3EZCW5/events.json","paper":"https://pith.science/paper/YTZH7TTL"},"agent_actions":{"view_html":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5","download_json":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5.json","view_paper":"https://pith.science/paper/YTZH7TTL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.1059&json=true","fetch_graph":"https://pith.science/api/pith-number/YTZH7TTLWAYNETVVR62D3EZCW5/graph.json","fetch_events":"https://pith.science/api/pith-number/YTZH7TTLWAYNETVVR62D3EZCW5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5/action/storage_attestation","attest_author":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5/action/author_attestation","sign_citation":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5/action/citation_signature","submit_replication":"https://pith.science/pith/YTZH7TTLWAYNETVVR62D3EZCW5/action/replication_record"}},"created_at":"2026-05-18T04:02:12.015852+00:00","updated_at":"2026-05-18T04:02:12.015852+00:00"}