{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "dilworth-finite-width-chain-decomposition-1950", "title": "Dilworth: finite-width chain decomposition of partially ordered sets", "summary": "A partially ordered set of finite width k—no antichain of size k+1 and an antichain of size k—can be partitioned into k disjoint chains, and fewer chains cannot suffice. Dilworth proves the finite case and extends it to arbitrary finite-width ordered sets using a maximal principle. The paper applies the decomposition to set representatives and finite distributive-lattice embeddings.", "boundary": "The infinite-set extension retains a finite integer width and uses the maximal principle; it is not an unrestricted infinite-cardinal chain-cover theorem or a constructive algorithm. The number of chains in a partition is not generally the Dushnik–Miller order dimension; the original distinguishes these representations. Hall/Rado and Kreweras representative theorems are inherited antecedents or consequences, not separately relabelled discoveries. The finite lattice application concerns lattice embeddings into a product of chains, not every non-distributive lattice.", "area": "Order theory and combinatorics", "topic": "Mathematics", "year": 1950, "lane": "HUMAN", "result": "THEOREM", "package_id": "dilworth-finite-width-chain-decomposition-1950", "target": { "record_id": "urn:uuid:15791703-9a7f-497a-a0b2-65a66b2e2930", "version": 1, "content_hash": "sha256:2ed9319acf1238bcfee9f33d6248a834a045d4c2ac39fb416251407afb03d418" }, "humans": [ "R. P. Dilworth" ], "authors_complete": true, "human_role_note": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. 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Dilworth proves the finite case and extends it to arbitrary finite-width ordered sets using a maximal principle. The paper applies the decomposition to set representatives and finite distributive-lattice embeddings.", "scope": "A partially ordered set of finite width k—no antichain of size k+1 and an antichain of size k—can be partitioned into k disjoint chains, and fewer chains cannot suffice. Dilworth proves the finite case and extends it to arbitrary finite-width ordered sets using a maximal principle. The paper applies the decomposition to set representatives and finite distributive-lattice embeddings. The infinite-set extension retains a finite integer width and uses the maximal principle; it is not an unrestricted infinite-cardinal chain-cover theorem or a constructive algorithm. The number of chains in a partition is not generally the Dushnik–Miller order dimension; the original distinguishes these representations. Hall/Rado and Kreweras representative theorems are inherited antecedents or consequences, not separately relabelled discoveries. 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Dilworth", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "prover" ], "source": { "citation": "A decomposition theorem for partially ordered sets (original1950)", "url": "https://daiwz.net/course/disc_math/2023/Dilworth.pdf" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "A decomposition theorem for partially ordered sets (original1950)", "url": "https://daiwz.net/course/disc_math/2023/Dilworth.pdf" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "The infinite-set extension retains a finite integer width and uses the maximal principle; it is not an unrestricted infinite-cardinal chain-cover theorem or a constructive algorithm. The number of chains in a partition is not generally the Dushnik–Miller order dimension; the original distinguishes these representations. Hall/Rado and Kreweras representative theorems are inherited antecedents or consequences, not separately relabelled discoveries. The finite lattice application concerns lattice embeddings into a product of chains, not every non-distributive lattice.", "domain": "Order theory and combinatorics" }, "sources": [ { "citation": "A decomposition theorem for partially ordered sets (original1950)", "url": "https://daiwz.net/course/disc_math/2023/Dilworth.pdf" } ], "statement": "# Dilworth: finite-width chain decomposition of partially ordered sets\n\nA partially ordered set of finite width k—no antichain of size k+1 and an antichain of size k—can be partitioned into k disjoint chains, and fewer chains cannot suffice. Dilworth proves the finite case and extends it to arbitrary finite-width ordered sets using a maximal principle. The paper applies the decomposition to set representatives and finite distributive-lattice embeddings.\n\n## Scientific scope\n\nThe infinite-set extension retains a finite integer width and uses the maximal principle; it is not an unrestricted infinite-cardinal chain-cover theorem or a constructive algorithm. The number of chains in a partition is not generally the Dushnik–Miller order dimension; the original distinguishes these representations. Hall/Rado and Kreweras representative theorems are inherited antecedents or consequences, not separately relabelled discoveries. The finite lattice application concerns lattice embeddings into a product of chains, not every non-distributive lattice.\n\n## Reading status\n\nThis is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.\n\n## People and systems\n\nR. P. Dilworth\n\nSource-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.\n\n\n\n## Original sources\n\n- [A decomposition theorem for partially ordered sets (original1950)](https://daiwz.net/course/disc_math/2023/Dilworth.pdf)\n", "summary": "A partially ordered set of finite width k—no antichain of size k+1 and an antichain of size k—can be partitioned into k disjoint chains, and fewer chains cannot suffice. Dilworth proves the finite case and extends it to arbitrary finite-width ordered sets using a maximal principle. The paper applies the decomposition to set representatives and finite distributive-lattice embeddings.", "title": "Dilworth: finite-width chain decomposition of partially ordered sets" }, "record_id": "urn:uuid:15791703-9a7f-497a-a0b2-65a66b2e2930", "version": 1, "source_ref": { "record_id": "urn:uuid:15791703-9a7f-497a-a0b2-65a66b2e2930", "version": 1, "content_hash": "sha256:2ed9319acf1238bcfee9f33d6248a834a045d4c2ac39fb416251407afb03d418" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:a9859ba4fc132dd5ae385b679e1164dbdb9ab60285574aa001677fa357248fda" }, "target": { "record_id": "urn:uuid:15791703-9a7f-497a-a0b2-65a66b2e2930", "version": 1, "content_hash": "sha256:2ed9319acf1238bcfee9f33d6248a834a045d4c2ac39fb416251407afb03d418" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "dilworth-finite-width-chain-decomposition-1950", "entry_id": "dilworth-finite-width-chain-decomposition-1950", "revision": 1, "title": "Dilworth: finite-width chain decomposition of partially ordered sets", "context": "Order theory and combinatorics", "track": "Exact theorem and reusable mathematical method", "band": "A", "modifier": "+", "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-23", "evidence_cutoff": "2026-10-23", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "A partially ordered set of finite width k—no antichain of size k+1 and an antichain of size k—can be partitioned into k disjoint chains, and fewer chains cannot suffice. Dilworth proves the finite case and extends it to arbitrary finite-width ordered sets using a maximal principle. The paper applies the decomposition to set representatives and finite distributive-lattice embeddings.", "scope": "A partially ordered set of finite width k—no antichain of size k+1 and an antichain of size k—can be partitioned into k disjoint chains, and fewer chains cannot suffice. Dilworth proves the finite case and extends it to arbitrary finite-width ordered sets using a maximal principle. The paper applies the decomposition to set representatives and finite distributive-lattice embeddings. The infinite-set extension retains a finite integer width and uses the maximal principle; it is not an unrestricted infinite-cardinal chain-cover theorem or a constructive algorithm. The number of chains in a partition is not generally the Dushnik–Miller order dimension; the original distinguishes these representations. Hall/Rado and Kreweras representative theorems are inherited antecedents or consequences, not separately relabelled discoveries. The finite lattice application concerns lattice embeddings into a product of chains, not every non-distributive lattice.", "evidence_summary": "The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The infinite-set extension retains a finite integer width and uses the maximal principle; it is not an unrestricted infinite-cardinal chain-cover theorem or a constructive algorithm. The number of chains in a partition is not generally the Dushnik–Miller order dimension; the original distinguishes these representations. Hall/Rado and Kreweras representative theorems are inherited antecedents or consequences, not separately relabelled discoveries. The finite lattice application concerns lattice embeddings into a product of chains, not every non-distributive lattice.", "references": [ { "title": "A decomposition theorem for partially ordered sets (original1950)", "url": "https://daiwz.net/course/disc_math/2023/Dilworth.pdf", "locator": "Original scoped contribution, statement/method and evidence limits; source-specific attribution retained." } ], "roles": [ { "who": "The named authors in the research record", "credit": "SCIENCE", "did": "Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred." } ], "target": { "record_id": "urn:uuid:15791703-9a7f-497a-a0b2-65a66b2e2930", "version": 1, "content_hash": "sha256:2ed9319acf1238bcfee9f33d6248a834a045d4c2ac39fb416251407afb03d418" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "245000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": null, "reading_status": "SOURCE_LINKED_NOTE" }