Ordered Sets

1. Front Matter

   Pages i-xviii

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2. Structure and Arithmetic of Ordered Sets

   1. Front Matter

      Pages 1-1

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   2. Arithmetic of Ordered Sets

      • Bjarni Jónsson

      Pages 3-41

   3. Exponentiation and Duality

      • B. A. Davey, D. Duffus

      Pages 43-95

   4. The Retract Construction

      • Ivan Rival

      Pages 97-122

3. Linear Extensions

   1. Front Matter

      Pages 123-123

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   2. Linear Extensions of Ordered Sets

      • R. Bonnet, M. Pouzet

      Pages 125-170

   3. Dimension Theory for Ordered Sets

      • David Kelly, William T. Trotter Jr.

      Pages 171-211

   4. Linear Extensions of Partial Orders and the FKG Inequality

      • R. L. Graham

      Pages 213-236

4. Set Theory and Recursion

   1. Front Matter

      Pages 237-237

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   2. Order Types of Real Numbers and Other Uncountable Orderings

      • James E. Baumgartner

      Pages 239-277

   3. On the Cofinality of Partially Ordered Sets

      • E. C. Milner, M. Pouzet

      Pages 279-298

   4. Infinite Ordered Sets, A Recursive Perspective

      • George F. McNulty

      Pages 299-330

5. Lattice Theory

   1. Front Matter

      Pages 331-331

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   2. The Role of Order in Lattice Theory

      • R. P. Dilworth

      Pages 333-353

   3. Some Order Theoretic Questions About Free Lattices and Free Modular Lattices

      • Ralph Freese

      Pages 355-376

   4. Addendum

      • Ivan Rival

      Pages 376-377

   5. An Introduction to the Theory of Continuous Lattices

      • Michael W. Mislove

      Pages 379-406

   6. Ordered Sets in Geometry

      • Garrett Birkhoff

      Pages 407-443

   7. Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts

      • Rudolf Wille

      Pages 445-470

This volume contains all twenty-three of the principal survey papers presented at the Symposium on Ordered Sets held at Banff, Canada from August 28 to September 12, 1981. The Symposium was supported by grants from the NATO Advanced Study Institute programme, the Natural Sciences and Engineering Research Council of Canada, the Canadian Mathematical Society Summer Research Institute programme, and the University of Calgary. tve are very grateful to these Organizations for their considerable interest and support. Over forty years ago on April 15, 1938 the first Symposium on Lattice Theory was held in Charlottesville, U.S.A. in conjunction with a meeting of the American Mathematical Society. The principal addresses on that occasion were Lattices and their applications by G. Birkhoff, On the application of structure theory to groups by O. Ore, and The representation of Boolean algebras by M. H. Stone. The texts of these addresses and three others by R. Baer, H. M. MacNeille, and K. Menger appear in the Bulletin of the American Mathematical Society, Volume 44, 1938. In those days the theory of ordered sets, and especially lattice theory was described as a "vigorous and promising younger brother of group theory." Some early workers hoped that lattice­ theoretic methods would lead to solutions of important problems in group theory.

• Lattice
• combinatorics
• computer
• computer science
• game theory
• geometry
• programming
• sets

• Department of Mathematics and Statistics, The University of Calgary, Canada

  Ivan Rival

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