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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 $\\mathcal{L}$"},"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":"2606.17274","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-06-15T20:33:31Z","cross_cats_sorted":[],"title_canon_sha256":"be851644d034fd4f43d2da4ae70999e95db67595d385959dcdc847284bc6b4ef","abstract_canon_sha256":"05f5eb22bd558bb487af07ec6167fb4958d43e53553adab964d0a18e198e914c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:10:07.869959Z","signature_b64":"yGHiL/0psS9pCK8Y5dbwyqrCScbnSe+BRizkEeKV3tcgd9+67aMX3KpVU6v4JoFZE+K8yIlqg0T8ZZK6YBLtAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91ba5d4677f2a13f8e57cb9360e1615c45b48164e3a0d6688fc23e231fed53f2","last_reissued_at":"2026-06-19T16:10:07.869608Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:10:07.869608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $h(x_\\downarrow)_\\mathcal{L} = h(x_\\downarrow)_{\\mathcal{L}'}$. 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