What is recorded here?
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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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.
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People and systems
R. P. Dilworth
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English catalogue narrative
Dilworth: finite-width chain decomposition of partially ordered sets
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
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.
Reading status
This 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.
People and systems
R. P. Dilworth
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.