{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:UJHFUCPHBEVQSNVZHBLAYZTHGL","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":"eba042dc2b402e576d350e15639f54a3246e92e334c8e1e79819e82b1076d25d","cross_cats_sorted":["math.AC","math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-05-20T17:13:17Z","title_canon_sha256":"e03d3b39c6187eb579fe54e0719da08ee22a7863e8b66a9baae714e6637e4f1c"},"schema_version":"1.0","source":{"id":"1305.4578","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1305.4578","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"arxiv_version","alias_value":"1305.4578v1","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.4578","created_at":"2026-05-18T01:04:41Z"},{"alias_kind":"pith_short_12","alias_value":"UJHFUCPHBEVQ","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"UJHFUCPHBEVQSNVZ","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"UJHFUCPH","created_at":"2026-05-18T12:28:02Z"}],"graph_snapshots":[{"event_id":"sha256:69ba75e085059deea7aad1d9060e82d705527459a962c2a4c47ec985ff525cf7","target":"graph","created_at":"2026-05-18T01:04:41Z","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 introduce the notion of a (strongly) topological lattice $\\mathcal{L}=(L,\\wedge ,\\vee)$ with respect to a subset $X\\subsetneqq L;$ aprototype is the lattice of (two-sided) ideals of a ring $R,$ which is(strongly) topological with respect to the prime spectrum of $R.$ We investigate and characterize (strongly) topological lattices. Given a non-zero left $R$-module $M,$ we introduce and investigate the spectrum $\\mathrm{Spec}^{\\mathrm{f}}(M)$ of \\textit{first submodules} of $M.$ We topologize $\\mathrm{Spec}^{\\mathrm{f}}(M)$ and investigate the algebraic properties of $_{R}M$ by passing to the","authors_text":"Christian Lomp, Jawad Abuhlail","cross_cats":["math.AC","math.GN"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-05-20T17:13:17Z","title":"On Topological Lattices and an Application to First Submodules"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.4578","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:f3b514020fe0718e64843e663400db741f46736f4e5bcb1fce69621d2c891c51","target":"record","created_at":"2026-05-18T01:04:41Z","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":"eba042dc2b402e576d350e15639f54a3246e92e334c8e1e79819e82b1076d25d","cross_cats_sorted":["math.AC","math.GN"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2013-05-20T17:13:17Z","title_canon_sha256":"e03d3b39c6187eb579fe54e0719da08ee22a7863e8b66a9baae714e6637e4f1c"},"schema_version":"1.0","source":{"id":"1305.4578","kind":"arxiv","version":1}},"canonical_sha256":"a24e5a09e7092b0936b938560c666732fa09095fbb989873326a833a01e71b9d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a24e5a09e7092b0936b938560c666732fa09095fbb989873326a833a01e71b9d","first_computed_at":"2026-05-18T01:04:41.913359Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:41.913359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qULF+4DjZWikRJFPq5+hgQ9T+K3Pcv9cxc41QulgYg9GuBucX85GBecz/gzb1JmUzJapdYmMifxpZaRakf7gCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:41.913852Z","signed_message":"canonical_sha256_bytes"},"source_id":"1305.4578","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f3b514020fe0718e64843e663400db741f46736f4e5bcb1fce69621d2c891c51","sha256:69ba75e085059deea7aad1d9060e82d705527459a962c2a4c47ec985ff525cf7"],"state_sha256":"fc6a1789860987e3e21e4adefb218d21e19bda1c9804816d0c64c7b718e717e2"}