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converse: For bounded orders $P$ and $Q$ and an isotone \\zo map $\\gy$ of $P$ into $Q$, we represent $P$ and $Q$ as $\\Princl K$ and $\\Princl L$ for bounded lattices $K$ and $L$ with a \\zo homomorphism $\\gf$ of $K$ into $L$, so that $\\gy$ is represented as $\\gf_{\\Hom}$."},"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":"1507.03270","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2015-07-12T20:32:02Z","cross_cats_sorted":[],"title_canon_sha256":"78999cbca4c3176a1166279b35d401e6145be359458dc20bce706025968d3604","abstract_canon_sha256":"d2ddb78830998affabed6302c2bf5dd2ecf7133f66847a25a17a79b30412771e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:15.733056Z","signature_b64":"MJNhIUjAy/XynKakOlUvqveQ3AZXKTKvX0cfNcnzJPbN8u4T2dwwlWvPKLCFTQNhIgNlr4FpEutiTanKxaVmDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8f97619ffa4a96db3891ded16d3107e688b59b1796cf5e63ec3b3bca7c249f9c","last_reissued_at":"2026-05-18T01:35:15.732514Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:15.732514Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homomorphisms and principal congruences of bounded lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"George Gr\\\"atzer","submitted_at":"2015-07-12T20:32:02Z","abstract_excerpt":"Two years ago, I characterized the order $\\Princl L$ of principal congruences of a bounded lattice $L$ as a bounded order.\n  If $K$ and $L$ are bounded lattices and $\\gf$ is a \\zo homomorphism of $K$ into~$L$, then there is a natural isotone \\zo-map $\\gf_{\\Hom}$ from $\\Princl K$ into $\\Princl L$.\n  We prove the converse: For bounded orders $P$ and $Q$ and an isotone \\zo map $\\gy$ of $P$ into $Q$, we represent $P$ and $Q$ as $\\Princl K$ and $\\Princl L$ for bounded lattices $K$ and $L$ with a \\zo homomorphism $\\gf$ of $K$ into $L$, so that $\\gy$ is represented as 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