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In this paper, we study some basic properties of the hom-poset $\\Hom(P,Q)$ and prove that $\\Hom\\big(P,J(Q)\\big)$ is a distributive lattice and characterized by \\[ \\Hom\\big(P,J(Q)\\big)\\cong J(P^*\\times Q), \\] where $P^*$ is the dual of $P$. Consequently, we obtain that $\\Hom\\big(P,J(Q)\\big)$ and $\\Hom\\big(Q,J(P)\\big)$ are dual isomorphic, i.e., \\[ \\Hom\\big(P,J(Q)\\big)\\cong \\Hom^{*}\\big(Q,J(P)\\big). \\] As applicatio"},"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":"1709.01234","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-09-05T04:27:24Z","cross_cats_sorted":[],"title_canon_sha256":"3f230e3645557973a96746338e4b3f90a9a84adb45119b0bd1c022c87b429c48","abstract_canon_sha256":"c705ca249b4315eb394e6addcc6ca385c911ed230eb563927762e257dd463e71"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:34.750622Z","signature_b64":"k7uRipgDQ3MGmbkr7Wk6c3FaGaVavGZn/U3PclNWHKoCQm7I89XvuB6qzPvbmCi6lGouhQCzbUS1MpbUnJdDDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1d6230a2609c55745d68b258a66992be249deecc825c3a5bfad32a003ad9ca89","last_reissued_at":"2026-05-18T00:21:34.749870Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:34.749870Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Order Preserving Maps of Posets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Suijie Wang, Zhousheng Mei","submitted_at":"2017-09-05T04:27:24Z","abstract_excerpt":"For any two finite posets $P$ and $Q$, let $\\Hom(P,Q)$ be the hom-poset consisting of all order preserving maps from $P$ to $Q$, and $J(Q)$ the collection of all order ideals of $Q$. In this paper, we study some basic properties of the hom-poset $\\Hom(P,Q)$ and prove that $\\Hom\\big(P,J(Q)\\big)$ is a distributive lattice and characterized by \\[ \\Hom\\big(P,J(Q)\\big)\\cong J(P^*\\times Q), \\] where $P^*$ is the dual of $P$. Consequently, we obtain that $\\Hom\\big(P,J(Q)\\big)$ and $\\Hom\\big(Q,J(P)\\big)$ are dual isomorphic, i.e., \\[ \\Hom\\big(P,J(Q)\\big)\\cong \\Hom^{*}\\big(Q,J(P)\\big). \\] As applicatio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.01234","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1709.01234","created_at":"2026-05-18T00:21:34.749989+00:00"},{"alias_kind":"arxiv_version","alias_value":"1709.01234v2","created_at":"2026-05-18T00:21:34.749989+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1709.01234","created_at":"2026-05-18T00:21:34.749989+00:00"},{"alias_kind":"pith_short_12","alias_value":"DVRDBITATRKX","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_16","alias_value":"DVRDBITATRKXIXLI","created_at":"2026-05-18T12:31:12.930513+00:00"},{"alias_kind":"pith_short_8","alias_value":"DVRDBITA","created_at":"2026-05-18T12:31:12.930513+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DVRDBITATRKXIXLIWJMKM2MSXY","json":"https://pith.science/pith/DVRDBITATRKXIXLIWJMKM2MSXY.json","graph_json":"https://pith.science/api/pith-number/DVRDBITATRKXIXLIWJMKM2MSXY/graph.json","events_json":"https://pith.science/api/pith-number/DVRDBITATRKXIXLIWJMKM2MSXY/events.json","paper":"https://pith.science/paper/DVRDBITA"},"agent_actions":{"view_html":"https://pith.science/pith/DVRDBITATRKXIXLIWJMKM2MSXY","download_json":"https://pith.science/pith/DVRDBITATRKXIXLIWJMKM2MSXY.json","view_paper":"https://pith.science/paper/DVRDBITA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1709.01234&json=true","fetch_graph":"https://pith.science/api/pith-number/DVRDBITATRKXIXLIWJMKM2MSXY/graph.json","fetch_events":"https://pith.science/api/pith-number/DVRDBITATRKXIXLIWJMKM2MSXY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DVRDBITATRKXIXLIWJMKM2MSXY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DVRDBITATRKXIXLIWJMKM2MSXY/action/storage_attestation","attest_author":"https://pith.science/pith/DVRDBITATRKXIXLIWJMKM2MSXY/action/author_attestation","sign_citation":"https://pith.science/pith/DVRDBITATRKXIXLIWJMKM2MSXY/action/citation_signature","submit_replication":"https://pith.science/pith/DVRDBITATRKXIXLIWJMKM2MSXY/action/replication_record"}},"created_at":"2026-05-18T00:21:34.749989+00:00","updated_at":"2026-05-18T00:21:34.749989+00:00"}