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Ordered Sets pp 555–581Cite as

The Möbius Function of a Partially Ordered Set

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

Abstract

The history of the Möbius function has many threads, involving aspects of number theory, algebra, geometry, topology, and combinatorics. The subject received considerable focus from Rota’s by now classic paper in which the Möbius function of a partially ordered set emerged in clear view as an important object of study. On the one hand, it can be viewed as an enumerative tool, defined implicitly by the relations

$$f(x) = \sum\limits_{yx} {g(y)} {\text{ and }}g(x) = \sum\limits_{yx} {\mu (y,x)f(y)} $$

where f and g are arbitrary functions on a poset P. On the other hand, one can study μ for its own sake as a combinatorial invariant giving important and useful information about the structure of P.

These expository lectures will trace this historical development and recent progress in the theory from both of these points of view. Topics to receive special attention are these:

  1. (i)

    the Möbius function as a geometric invariant, for geometric lattices and other ordered structures associated with geometries;

  2. (ii)

    algebraic and homological methods;

  3. (iii)

    a catalog of interesting families of partially ordered sets for which the Möbius function is known.

This paper is a brief expository account of some basic results in the theory of Möbius functions on partially ordered sets. It is not a survey, and no attempt will be made to be complete. What this paper represents is a summary, with complete proofs, of the basic results upon which the subject rests. For the most part, these are taken from Rotafs pivotal paper, “On the foundations of combinatorial theory I: The theory of Möbius functions” [Ro1], and from several papers of Crapo ([Crl], [Cr2]), in which Rota’s work was extended significantly. We have endeavored to refine and condense the proofs as much as possible, although this has also led us to abandon interesting (but less efficient) lines of development. For the reader interested in learning more, there is a substantial bibliography including many recent papers of considerable importance.

If there is a “modern era” of the Möbius function, it begins in 1964 with Rota’s paper [Rol]. One could argue, of course, that the story begins earlier: the Möbius function of elementary number theory has a rich and varied history stretching back to the last century. The inclusion-exclusion principle has its roots in combinatorial antiquity. Significant steps were taken before 1964 (e.g. by Weisner, P. Hall, Dilworth, and others) toward developing and using a theory of Möbius inversion on arbitrary partially ordered sets. However Rota’s paper marks the time at which the Möbius function emerged in clear view as a fundamental invariant, which unifies both enumerative and structural aspects of the theory of partially ordered sets. The title of [Rol] is a startling prophesy, fulfilled to a great extent by the paper itself, and also by many lines of research which continue to this day.

We hope this paper will help whet the appetite of those not yet fully acquainted with this interesting and important subject.

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Authors and Affiliations

  1. Department of Mathematics, Haverford College, Haverford, Pennsylvania, 19041, USA

    Curtis Greene

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  1. Curtis Greene
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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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Greene, C. (1982). The Möbius Function of a Partially Ordered Set. 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_18

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  • DOI: https://doi.org/10.1007/978-94-009-7798-3_18

  • Publisher Name: Springer, Dordrecht

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