Advertisement

Incidence Functions

  • Martin Aigner
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 234)

Abstract

After having computed the level numbers and the total number of elements of some important lattices in the last chapter we now consider counting functions and inversion formulae in an arbitrary poset. Our method of study will be to associate with the poset P an algebraic object called the incidence algebra A (P), and to investigate its structure and subobjects.

Keywords

Rank Function Inversion Formula Maximal Chain Irreducible Element Finite Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

  1. 1.
    Ward, M. Arithmetic functions on rings. Ann. Math. 38, 725–732 (1937).CrossRefGoogle Scholar
  2. 2.
    Ward, M. The algebra of lattice functions. Duke Math. J. 5, 357–371 (1939).Google Scholar
  3. 1.
    Weisner, L. Abstract theory of inversion of finite series. Trans. Amer. Math. Soc. 38, 474–484 (1935).MathSciNetGoogle Scholar
  4. 4.
    Carlitz, L. Rings of arithmetic functions. Pac. J. Math. 14, 1165–1171 (1964).zbMATHGoogle Scholar
  5. 1.
    Rota, G.-C.-Smith, D. A. Fluctuation theory and Baxter algebras. Symp. Math. Ist. Naz. Alta Mat. 9, 179–201 (1972).Google Scholar
  6. 1.
    J. P. S.-Rota, G.-C. Invariant theory, Young bitableaux, and Combinatorics. Advances Math. 27, 63–92 (1978).MathSciNetzbMATHGoogle Scholar
  7. 1.
    Mullin, R. On Rota’s problem concerning partitions. Aequationes Math. 2, 98–104 (1969).Google Scholar
  8. 2.
    Crapo, H. H. The Möbius function of a lattice. J. Comb. Theory 1, 126–131 (1966).MathSciNetzbMATHCrossRefGoogle Scholar
  9. 5.
    Crapo, H. H. Möbius inversion in lattices. Archiv Math. 19, 595–607 (1968).MathSciNetGoogle Scholar
  10. 1.
    Comtet, L. Advanced Combinatorics. Dordrecht and Boston: Reidel (1974).Google Scholar
  11. 1.
    Ryser, H. J. Combinatorial Mathematics. Carus Math. Monographs Nr. 14, New York: Math. Ass. Amer. (1963).Google Scholar
  12. 2.
    Ryser, H. J. A combinatorial theorem with applications to Latin rectangles. Proc. Amer. Math. Soc. 2, 550¬552 (1951).Google Scholar
  13. 1.
    Riordan, J. An Introduction to Combinatorial Mathematics. New York, London, Sydney: John Wiley & Sons (1958).Google Scholar
  14. 2.
    Riordan, J. Combinatorial Identities. New York, London, Sidney: John Wiley & Sons (1968).Google Scholar
  15. 4.
    Greene, C. On the Möbius algebra of a partially ordered set. Advances Math. 10, 177–187 (1973).MathSciNetzbMATHGoogle Scholar
  16. 4.
    Rota, G.-C. On the combinatorics of the Euler characteristic. Studies in Pure Math. (Mirsky, ed)., 221–233. London: Academic Press (1971).Google Scholar
  17. 2.
    Davis, R. L. Order algebras. Bull. Amer. Math. Soc. 76, 83–87 (1970).zbMATHGoogle Scholar
  18. 1.
    Geissinger, L. Valuations on distributive lattices I, II, III. Archiv Math. 24, 230–239; 337–345; 475–481 (1973).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1979

Authors and Affiliations

  • Martin Aigner
    • 1
  1. 1.II. Institut für MathematikFreie Universität BerlinBerlin 33Federal Republic of Germany

Personalised recommendations