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An Introduction to Cohen-Macaulay Partially Ordered Sets

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Ordered Sets

Part of the book series: NATO Advanced Study Institutes Series ((ASIC,volume 83))

Abstract

Combinatorics, algebra and topology come together in a most remarkable way in the theory of Cohen-Macaulay posets. These lectures will provide an introduction to the subject based on the work of Baclawski, Hochster, Reisner and the present authors (see references).

Combinatorial properties of some important posets will be surveyed. This leads in a natural way to the concept of a Cohen- Macaulay poset. This concept can be formulated in terms of a certain ring associated with the. poset. On the other hand, Cohen-Macaulay posets can be defined in terms of topological properties. A fundamental theorem of Reisner gives the connection between these two definitions. Examples of Cohen-Macaulay posets include semmodular lattices (in particular, distributive and geometric lattices), supersolvable lattices, face-lattices of polytopes, and Bruhat order. This theory has given birth to the concept of lexicographically shellable posets. Their ubiquity and usefulness give them a central position in the subject.

The theory of Cohen-Macaulay posets has many applications both to combinatorics and to other branches of mathematics, including algebra and topology. It can be used, for instance, to obtain Information about such numerical invariants of posets as number of chains and Möbius function. Cohen-Macaulay posets can be used to study rings of interest to algebraic geometry and establish the topological type of certain simplicial complexes. Contributions to representation theory can be made by considering groups acting on Cohen-Macaulay posets. Examples of these and other applications will be discussed, together with open problems.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Stockholm, Box 6701, S-113 85, Stockholm, Sweden

    A. Björner

  2. Department of Mathematics, University of California — San Diego, La Jolla, California, 92093, USA

    A. M. Garsia

  3. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA

    R. P. Stanley

Authors
  1. A. Björner
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  2. A. M. Garsia
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  3. R. P. Stanley
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Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, The University of Calgary, Canada

    Ivan Rival

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© 1982 D. Reidel Publishing Company

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Björner, A., Garsia, A.M., Stanley, R.P. (1982). An Introduction to Cohen-Macaulay Partially Ordered Sets. In: Rival, I. (eds) Ordered Sets. NATO Advanced Study Institutes Series, vol 83. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7798-3_19

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  • DOI: https://doi.org/10.1007/978-94-009-7798-3_19

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-7800-3

  • Online ISBN: 978-94-009-7798-3

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