{"paper":{"title":"Congruences and trajectories in planar semimodular lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"George Gr\\\"atzer","submitted_at":"2014-06-02T16:40:59Z","abstract_excerpt":"A 1955 result of J.~Jakub\\'i k states that for the prime intervals $\\fp$ and $\\fq$ of a finite lattice, $\\con{\\fp} \\geq \\con{\\fq}$ if{}f $\\fp$ is congruence-projective to~$\\fq$ (\\emph{via} intervals of arbitrary size). The problem is how to determine whether $\\con{\\fp} \\geq \\con{\\fq}$ involving only prime intervals.\n  Two recent papers approached this problem in different ways. G. Cz\\'edli's used trajectories for slim rectangular lattices---a special subclass of slim, planar, semimodular lattices. I used the concept of prime-projectivity for arbitrary finite lattices. In this note I show how m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.0439","kind":"arxiv","version":4},"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"}