The Dilworth Theorems: Selected Papers of Robert P. Dilworth

Front Matter....Pages i-xxvi
Front Matter....Pages 1-6
A Decomposition Theorem for Partially Ordered Sets....Pages 7-12
Some Combinatorial Problems on Partially Ordered Sets....Pages 13-18
The Impact of the Chain Decomposition Theorem on Classical Combinatorics....Pages 19-29
Dilworth’s Decomposition Theorem in the Infinite Case....Pages 30-35
Effective Versions of the Chain Decomposition Theorem....Pages 36-38
Front Matter....Pages 39-40
Lattices with Unique Complements....Pages 41-72
On Complemented Lattices....Pages 73-78
Uniquely Complemented Lattices....Pages 79-84
On Orthomodular Lattices....Pages 85-87
Front Matter....Pages 89-92
Lattices with Unique Irreducible Decompositions....Pages 93-99
The Arithmetical Theory of Birkhoff Lattices....Pages 101-114
Ideals in Birkhoff Lattices....Pages 115-143
Decomposition Theory for Lattices without Chain Conditions....Pages 145-166
Note on the Kurosch-Ore Theorem....Pages 167-171
Structure and Decomposition Theory of Lattices....Pages 173-186
Dilworth’s Work on Decompositions in Semimodular Lattices....Pages 187-191
The Consequences of Dilworth’s Work on Lattices with Unique Irreducible Decompositions....Pages 192-199
Exchange Properties for Reduced Decompositions in Modular Lattices....Pages 200-202
The Impact of Dilworth’s Work on Semimodular Lattices on the Kurosch-Ore Theorem....Pages 203-204
Front Matter....Pages 205-209
The Imbedding Problem for Modular Lattices....Pages 211-217
Front Matter....Pages 205-209
Proof of a Conjecture on Finite Modular Lattices....Pages 219-224
Distributivity in Lattices....Pages 225-235
Aspects of distributivity....Pages 237-250
The Role of Gluing Constructions in Modular Lattice Theory....Pages 251-260
Dilworth’s Covering Theorem for Modular Lattices....Pages 261-264
Front Matter....Pages 265-267
Dependence Relations in a Semi-Modular Lattice....Pages 269-281
A Counterexample to the Generalization of Sperner’s Theorem....Pages 283-286
Dilworth’s Completion, Submodular Functions, and Combinatorial Optimization....Pages 287-294
Dilworth Truncations of Geometric Lattices....Pages 295-297
The Sperner Property in Geometric and Partition Lattices....Pages 298-304
Front Matter....Pages 305-307
Abstract Residuation over Lattices....Pages 309-315
Residuated Lattices....Pages 317-336
Non-Commutative Residuated Lattices....Pages 337-355
Non-Commutative Arithmetic....Pages 357-367
Abstract Commutative Ideal Theory....Pages 369-386
Dilworth’s Early Papers on Residuated and Multiplicative Lattices....Pages 387-390
Abstract Ideal Theory: Principals and Particulars....Pages 391-396
Representation and Embedding Theorems for Noether Lattices and r -Lattices....Pages 397-402
Front Matter....Pages 403-405
The Structure of Relatively Complemented Lattices....Pages 407-418
The Normal Completion of the Lattice of Continuous Functions....Pages 419-430
Front Matter....Pages 403-405
A Generalized Cantor Theorem....Pages 431-432
Generators of lattice varieties....Pages 433-437
Lattice Congruences and Dilworth’s Decomposition of Relatively Complemented Lattices....Pages 439-444
The Normal Completion of the Lattice of Continuous Functions....Pages 445-449
Cantor Theorems for Relations....Pages 450-450
Ideal and Filter Constructions in Lattice Varieties....Pages 451-453
Front Matter....Pages 455-457
Dilworth’s Proof of the Embedding Theorem....Pages 458-459
On the Congruence Lattice of a Lattice....Pages 460-464
Back Matter....Pages 465-465