{"paper":{"title":"The M\\_{3}[D] construction and n-modularity","license":"","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Friedrich Wehrung (LMNO), George Gr\\\"atzer","submitted_at":"2005-01-25T08:22:31Z","abstract_excerpt":"In 1968, E. T. Schmidt introduced the M\\_3[D] construction, an extension of the five-element nondistributive lattice M\\_3 by a bounded distributive lattice D, defined as the lattice of all triples $(x, y, z) \\in D^3$ satisfying $x \\mm y = x \\mm z = y \\mm z$. The lattice M\\_3[D] is a modular congruence-preserving extension of D. In this paper, we investigate this construction for an arbitrary lattice L. For every n > 0, we exhibit an identity Un such that U1 is modularity and Un+1 is properly weaker than Un. Let Mn denote the variety defined by Un, the variety of n-modular lat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0501430","kind":"arxiv","version":1},"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"}