The Dilworth theorems. Selected papers of Robert P. Dilworth. Edited by Kenneth P. Bogart, Ralph Freese and Joseph P. S. Kung.

*(English)*Zbl 0907.06001
Contemporary Mathematicians. Boston, MA: Birkhäuser. xxvi, 465 p. (1990).

Publisher’s description: “This volume contains almost all of the fundamental mathematical papers of Robert P. Dilworth, one of the most important pioneers in lattice theory, combinatorics, and universal algebra. His incisive, specific results have served – and are still serving – as landmarks and inspirations for further work. Included in this volume are Dilworth’s papers on chain decompositions in ordered sets, lattices with unique complements, decomposition theory, completions of submodular functions, distributivity, and covering equalities in modular lattices.

These carefully selected papers directly display the distinctive qualities of Dilworth’s arguments. Dilworth has written backgrounds to many of the papers that yield much insight into his way of thinking and his strategy for solving problems. All the papers are provided with commentaries by leading experts that trace the influence of Dilworth’s ideas in later work, using them as focal points for mini-surveys of the field. Extensive bibliographies are included.

Because Dilworth’s work formed the starting point for many key developments in the theory of lattices and ordered sets, the book can be used as an introduction to these areas, accessible to students and at the same time useful to experts. It will also be of interest to historians of mathematics as a documentary on the early history of lattice theory.”

Contents: Editors’ preface (pp. xi–xiv); Biography (pp. xv–xvii); Peter Crawley, “Recollections of R. P. Dilworth” (pp. xix–xx); Phillip J. Chase, “Recollections of Professor Dilworth” (pp. xxi–xxii); Mathematical publications of Robert P. Dilworth (pp. xxiii–xxv); Doctoral students (p. xxvi).

Chapter 1. Chain partitions in ordered sets: R. P. Dilworth, “A decomposition theorem for partially ordered sets” [Zbl 0038.02003] (pp. 7-12); R. P. Dilworth, “Some combinatorial problems on partially ordered sets” [Zbl 0096.00601] (pp. 13-18); Kenneth P. Bogart, Curtis Greene and Joseph P. S. Kung, “The impact of the chain decomposition theorem on classical combinatorics” (pp. 19-29); E. C. Milner, “Dilworth’s decomposition theorem in the infinite case” (pp. 30-35); H. Kierstead, “Effective versions of the chain decomposition theorem” (pp. 36-38).

Chapter 2. Complementation: R. P. Dilworth, “Lattices with unique complements” [Zbl 0060.06103] (pp. 41-72); R. P. Dilworth, “On complemented lattices” [Zbl 0023.10202] (pp. 73-78); M. E. Adams, “Uniquely complemented lattices” (pp. 79-84); Gudrun Kalmbach, “On orthomodular lattices” (pp. 85-87).

Chapter 3. Decomposition theory: R. P. Dilworth, “Lattices with unique irreducible decompositions” [Zbl 0025.10202] (pp. 93-99); R. P. Dilworth, “The arithmetical theory of Birkhoff lattices” [Zbl 0025.10203] (pp. 101-114); R. P. Dilworth, “Ideals in Birkhoff lattices” [Zbl 0025.01203] (pp. 115-143); R. P. Dilworth and Peter Crawley, “Decomposition theory for lattices without chain conditions” [Zbl 0091.03004] (pp. 145-166); R. P. Dilworth, “Note on the Kurosch-Ore theorem” [Zbl 0060.06107] (pp. 167-171); R. P. Dilworth, “Structure and decomposition theory of lattices” [Zbl 0101.27004] (pp. 173-186); Bjarni Jonsson, “Dilworth’s work on decomposition in semimodular lattices” (pp. 187-191); Bernard Monjardet, “The consequences of Dilworth’s work on lattices with unique irreducible decompositions” (pp. 192-199); Joseph P. S. Kung, “Exchange properties for reduced decompositions in modular lattices” (pp. 200-202); Manfred Stern, “The impact of Dilworth’s work on semimodular lattices on the Kurosch-Ore theorem” (pp. 203-204).

Chapter 4. Modular and distributive lattices: M. Hall and R. P. Dilworth, “The imbedding problem for modular lattices” [Zbl 0060.06102] (pp. 211-217); R. P. Dilworth, “Proof of a conjecture on finite modular lattices” [Zbl 0056.26203] (pp. 219-224); R. P. Dilworth and J. E. McLaughlin, “Distributivity in lattices” [Zbl 0047.26103] (pp. 225-235); R. P. Dilworth, “Aspects of distributivity” [Zbl 0541.06005] (pp. 237-250); Alan Day and Ralph Freese, “The role of gluing constructions in modular lattice theory” (pp. 251-260); Ivan Rival, “Dilworth’s covering theorem for modular lattices” (pp. 261-264).

Chapter 5. Geometric and semimodular lattices: R. P. Dilworth, “Dependence relations in a semimodular lattice” [Zbl 0060.06101] (pp. 269-281); R. P. Dilworth and Curtis Greene, “A counterexample to the generalization of Sperner’s theorem” [Zbl 0257.05022] (pp. 283-286); Ulrich Faigle, “Dilworth’s completion, submodular functions, and combinatorial optimization” (pp. 287-294); Joseph P. S. Kung, “Dilworth truncations of geometric lattices” (pp. 295-297); Jerrold R. Griggs, “The Sperner property in geometric and partition lattices” (pp. 298-304).

Chapter 6. Multiplicative lattices: R. P. Dilworth, “Abstract residuation over lattices” [Zbl 0018.34104] (pp. 309-315); Morgan Ward and R. P. Dilworth, “Residuated lattices” [Zbl 0021.10801] (pp. 317-336); R. P. Dilworth, “Noncommutative residuated lattices” [Zbl 0022.10402] (pp. 337-355); R. P. Dilworth, “Noncommutative arithmetic” [Zbl 0021.10704] (pp. 357-367); R. P. Dilworth, “Abstract commutative ideal theory” [Zbl 0111.04104] (pp. 369-386); D. D. Anderson, “Dilworth’s early papers on residuated and multiplicative lattices” (pp. 387-390); E. W. Johnson, “Abstract ideal theory: principals and particulars” (pp. 391-396); D. D. Anderson, “Representation and embedding theorems for Noether lattices and \(r\)-lattices” (pp. 397-402).

Chapter 7. Miscellaneous papers: R. P. Dilworth, “The structure of relatively complemented lattices” [Zbl 0036.01802] (pp. 407-418); R. P. Dilworth, “The normal completion of the lattice of continuous functions” [Zbl 0037.20205] (pp. 419-430); A. M. Gleason and R. P. Dilworth, “A generalized Cantor theorem” [Zbl 0109.24203] (pp. 431-432); R. P. Dilworth and Ralph Freese, “Generators of lattice varieties” [Zbl 0381.06008] (pp. 433-437); George F. McNulty, “Lattice congruences and Dilworth’s decomposition of relatively complemented lattices” (pp. 439-444); Gerhard Gierz, “The normal completion of the lattice of continuous functions” (pp. 445-449); Joseph P. S. Kung, “Cantor theorems for relations” (p. 450); J. B. Nation, “Ideal and filter constructions in lattice varieties” (pp. 451-453).

Chapter 8, Two results from “Algebraic theory of lattices”: Joseph P. S. Kung, “Dilworth’s proof of the embedding theorem” (pp. 458-459); George Grätzer, “On the congruence lattice of a lattice” (pp. 460-464).

These carefully selected papers directly display the distinctive qualities of Dilworth’s arguments. Dilworth has written backgrounds to many of the papers that yield much insight into his way of thinking and his strategy for solving problems. All the papers are provided with commentaries by leading experts that trace the influence of Dilworth’s ideas in later work, using them as focal points for mini-surveys of the field. Extensive bibliographies are included.

Because Dilworth’s work formed the starting point for many key developments in the theory of lattices and ordered sets, the book can be used as an introduction to these areas, accessible to students and at the same time useful to experts. It will also be of interest to historians of mathematics as a documentary on the early history of lattice theory.”

Contents: Editors’ preface (pp. xi–xiv); Biography (pp. xv–xvii); Peter Crawley, “Recollections of R. P. Dilworth” (pp. xix–xx); Phillip J. Chase, “Recollections of Professor Dilworth” (pp. xxi–xxii); Mathematical publications of Robert P. Dilworth (pp. xxiii–xxv); Doctoral students (p. xxvi).

Chapter 1. Chain partitions in ordered sets: R. P. Dilworth, “A decomposition theorem for partially ordered sets” [Zbl 0038.02003] (pp. 7-12); R. P. Dilworth, “Some combinatorial problems on partially ordered sets” [Zbl 0096.00601] (pp. 13-18); Kenneth P. Bogart, Curtis Greene and Joseph P. S. Kung, “The impact of the chain decomposition theorem on classical combinatorics” (pp. 19-29); E. C. Milner, “Dilworth’s decomposition theorem in the infinite case” (pp. 30-35); H. Kierstead, “Effective versions of the chain decomposition theorem” (pp. 36-38).

Chapter 2. Complementation: R. P. Dilworth, “Lattices with unique complements” [Zbl 0060.06103] (pp. 41-72); R. P. Dilworth, “On complemented lattices” [Zbl 0023.10202] (pp. 73-78); M. E. Adams, “Uniquely complemented lattices” (pp. 79-84); Gudrun Kalmbach, “On orthomodular lattices” (pp. 85-87).

Chapter 3. Decomposition theory: R. P. Dilworth, “Lattices with unique irreducible decompositions” [Zbl 0025.10202] (pp. 93-99); R. P. Dilworth, “The arithmetical theory of Birkhoff lattices” [Zbl 0025.10203] (pp. 101-114); R. P. Dilworth, “Ideals in Birkhoff lattices” [Zbl 0025.01203] (pp. 115-143); R. P. Dilworth and Peter Crawley, “Decomposition theory for lattices without chain conditions” [Zbl 0091.03004] (pp. 145-166); R. P. Dilworth, “Note on the Kurosch-Ore theorem” [Zbl 0060.06107] (pp. 167-171); R. P. Dilworth, “Structure and decomposition theory of lattices” [Zbl 0101.27004] (pp. 173-186); Bjarni Jonsson, “Dilworth’s work on decomposition in semimodular lattices” (pp. 187-191); Bernard Monjardet, “The consequences of Dilworth’s work on lattices with unique irreducible decompositions” (pp. 192-199); Joseph P. S. Kung, “Exchange properties for reduced decompositions in modular lattices” (pp. 200-202); Manfred Stern, “The impact of Dilworth’s work on semimodular lattices on the Kurosch-Ore theorem” (pp. 203-204).

Chapter 4. Modular and distributive lattices: M. Hall and R. P. Dilworth, “The imbedding problem for modular lattices” [Zbl 0060.06102] (pp. 211-217); R. P. Dilworth, “Proof of a conjecture on finite modular lattices” [Zbl 0056.26203] (pp. 219-224); R. P. Dilworth and J. E. McLaughlin, “Distributivity in lattices” [Zbl 0047.26103] (pp. 225-235); R. P. Dilworth, “Aspects of distributivity” [Zbl 0541.06005] (pp. 237-250); Alan Day and Ralph Freese, “The role of gluing constructions in modular lattice theory” (pp. 251-260); Ivan Rival, “Dilworth’s covering theorem for modular lattices” (pp. 261-264).

Chapter 5. Geometric and semimodular lattices: R. P. Dilworth, “Dependence relations in a semimodular lattice” [Zbl 0060.06101] (pp. 269-281); R. P. Dilworth and Curtis Greene, “A counterexample to the generalization of Sperner’s theorem” [Zbl 0257.05022] (pp. 283-286); Ulrich Faigle, “Dilworth’s completion, submodular functions, and combinatorial optimization” (pp. 287-294); Joseph P. S. Kung, “Dilworth truncations of geometric lattices” (pp. 295-297); Jerrold R. Griggs, “The Sperner property in geometric and partition lattices” (pp. 298-304).

Chapter 6. Multiplicative lattices: R. P. Dilworth, “Abstract residuation over lattices” [Zbl 0018.34104] (pp. 309-315); Morgan Ward and R. P. Dilworth, “Residuated lattices” [Zbl 0021.10801] (pp. 317-336); R. P. Dilworth, “Noncommutative residuated lattices” [Zbl 0022.10402] (pp. 337-355); R. P. Dilworth, “Noncommutative arithmetic” [Zbl 0021.10704] (pp. 357-367); R. P. Dilworth, “Abstract commutative ideal theory” [Zbl 0111.04104] (pp. 369-386); D. D. Anderson, “Dilworth’s early papers on residuated and multiplicative lattices” (pp. 387-390); E. W. Johnson, “Abstract ideal theory: principals and particulars” (pp. 391-396); D. D. Anderson, “Representation and embedding theorems for Noether lattices and \(r\)-lattices” (pp. 397-402).

Chapter 7. Miscellaneous papers: R. P. Dilworth, “The structure of relatively complemented lattices” [Zbl 0036.01802] (pp. 407-418); R. P. Dilworth, “The normal completion of the lattice of continuous functions” [Zbl 0037.20205] (pp. 419-430); A. M. Gleason and R. P. Dilworth, “A generalized Cantor theorem” [Zbl 0109.24203] (pp. 431-432); R. P. Dilworth and Ralph Freese, “Generators of lattice varieties” [Zbl 0381.06008] (pp. 433-437); George F. McNulty, “Lattice congruences and Dilworth’s decomposition of relatively complemented lattices” (pp. 439-444); Gerhard Gierz, “The normal completion of the lattice of continuous functions” (pp. 445-449); Joseph P. S. Kung, “Cantor theorems for relations” (p. 450); J. B. Nation, “Ideal and filter constructions in lattice varieties” (pp. 451-453).

Chapter 8, Two results from “Algebraic theory of lattices”: Joseph P. S. Kung, “Dilworth’s proof of the embedding theorem” (pp. 458-459); George Grätzer, “On the congruence lattice of a lattice” (pp. 460-464).

##### MSC:

06-02 | Research exposition (monographs, survey articles) pertaining to ordered structures |

01A75 | Collected or selected works; reprintings or translations of classics |

06-06 | Proceedings, conferences, collections, etc. pertaining to ordered structures |

01-06 | Proceedings, conferences, collections, etc. pertaining to history and biography |