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CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be the intersection of jCRL and mCRL. We prove CRL is a strict subset of biCRL which is a strict subset of both jCRL and mCRL. Let L be a DL. Then L is in mCRL iff L has a distinguishing set of comple"},"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":"1201.2331","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2012-01-11T15:30:22Z","cross_cats_sorted":[],"title_canon_sha256":"1c29fb3bf1fe89cd42ad8535fea60ad93978bc54dc5605e634fcfdb2f906776e","abstract_canon_sha256":"bd8e5673c0da93e2c2fdb3fd0953e98c9ce84a6f93813f3aae7597682ab2647f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:21.435213Z","signature_b64":"laMqKJHDbrRg9XDJSt7bV8vzFfpGOLxFYN+fwAl8UTCUDHiOPRVRuy3R5Mw89EVmzzKR7Xe40T7GpR7EliBwCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aab7cb1ad3552694ec788ab0e34397f74f1a45593e7d03695588fd056cecff51","last_reissued_at":"2026-05-18T01:07:21.434804Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:21.434804Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Completely Representable Lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Robert Egrot, Robin Hirsch","submitted_at":"2012-01-11T15:30:22Z","abstract_excerpt":"It is known that a lattice is representable as a ring of sets iff the lattice is distributive. 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