AWhat 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.
Scientific 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
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.
Contributors
- The named authors in the research record · SCIENCE: 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.
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