{"paper":{"title":"On some posets and lattices with the same height","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hoan La","submitted_at":"2026-06-15T20:33:31Z","abstract_excerpt":"For a finite poset $\\mathcal{P}$, its height $h(\\mathcal{P})$ is the number of cover relations in its longest chain. When $\\mathcal{P}$ is a lattice $\\mathcal{L}$, we label its elements $x$ with $h(x_\\downarrow) = h([\\hat{0},x])$ and its cover relations $x \\lessdot y$ with $h(y_\\downarrow) - h(x_\\downarrow)$. When a lattice $\\mathcal{L}'$ extends $\\mathcal{L}$, $h(x_\\downarrow)_\\mathcal{L} \\leq h(x_\\downarrow)_{\\mathcal{L}'}$. We study lattices $\\mathcal{L}$ and $\\mathcal{L}'$ such that $h(x_\\downarrow)_\\mathcal{L} = h(x_\\downarrow)_{\\mathcal{L}'}$. Cover relations labeled $1$ in $\\mathcal{L}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.17274","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.17274/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}