{"paper":{"title":"Varieties and quasivarieties of lattices with complementation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"A semidirect-product-like construction creates infinitely many finite subdirectly irreducible modular lattices with complementation satisfying a quasi-identity.","cross_cats":[],"primary_cat":"math.RA","authors_text":"H. L\\\"anger, I. Chajda, J. K\\\"uhr, V. Cenker","submitted_at":"2026-05-17T05:52:31Z","abstract_excerpt":"We investigate (quasi)varieties of lattices with complementation, i.e., complemented lattices equipped with a fixed complementation as a unary operation. We focus on subclasses satisfying additional conditions, such as the quasi-identity $({x'\\wedge y\\approx 0} \\;\\&\\; {x\\wedge y'\\approx 0})$ $\\Rightarrow x\\approx y$, modularity, or De Morgan's laws. We present a construction resembling a semidirect product that yields infinitely many finite subdirectly irreducible modular lattices with complementation satisfying this quasi-identity. We axiomatize small varieties, each of which covers the varie"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"A construction resembling a semidirect product yields infinitely many finite subdirectly irreducible modular lattices with complementation satisfying the quasi-identity (x'∧y≈0 & x∧y'≈0) ⇒ x≈y.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The semidirect-product-like construction produces structures that remain lattices, are modular, and satisfy the stated quasi-identity for the chosen complementation operation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Presents a construction for infinitely many finite subdirectly irreducible modular lattices with complementation satisfying a quasi-identity and axiomatizes small covering varieties generated by small modular De Morgan lattices.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"A semidirect-product-like construction creates infinitely many finite subdirectly irreducible modular lattices with complementation satisfying a quasi-identity.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"426410f199d0b0b9b0c4816e997bd4dfced09f7cafd500180e3fdf76cd836de0"},"source":{"id":"2605.17274","kind":"arxiv","version":1},"verdict":{"id":"144de905-1504-4e39-bcba-b7e4ab5e74a4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:59:41.277220Z","strongest_claim":"A construction resembling a semidirect product yields infinitely many finite subdirectly irreducible modular lattices with complementation satisfying the quasi-identity (x'∧y≈0 & x∧y'≈0) ⇒ x≈y.","one_line_summary":"Presents a construction for infinitely many finite subdirectly irreducible modular lattices with complementation satisfying a quasi-identity and axiomatizes small covering varieties generated by small modular De Morgan lattices.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The semidirect-product-like construction produces structures that remain lattices, are modular, and satisfy the stated quasi-identity for the chosen complementation operation.","pith_extraction_headline":"A semidirect-product-like construction creates infinitely many finite subdirectly irreducible modular lattices with complementation satisfying a quasi-identity."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17274/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T23:31:20.201868Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T23:13:20.951251Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T22:01:57.833559Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.776276Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"f7a55bc4b04398dba178d1ac94aa8332a372529bfc0c020db347e8c5b836a7f1"},"references":{"count":14,"sample":[{"doi":"","year":2005,"title":"In: Kudryavtsev, V.B., Rosenberg, I.G., Goldstein, M","work_id":"79f18652-9bd0-4fcf-8075-36ac13a0563f","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1979,"title":"AMS, Providence (1979) 30 V","work_id":"0a5e56e3-05ee-4bde-a811-eb568942df56","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"Blyth, T.S.: Lattices and Ordered Algebraic Structures. 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