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BIT Numerical Mathematics

, Volume 9, Issue 2, pp 97–118 | Cite as

Algorithms over partially ordered sets

  • R. M. Baer
  • O. Østerby
Article

Abstract

We here study some problems concerned with the computational analysis of finite partially ordered sets. We begin (in § 1) by showing that the matrix representation of a binary relationR may always be taken in triangular form ifR is a partial ordering. We consider (in § 2) the chain structure in partially ordered sets, answer the combinatorial question of how many maximal chains might exist in a partially ordered set withn elements, and we give an algorithm for enumerating all maximal chains. We give (in § 3) algorithms which decide whether a partially ordered set is a (lower or upper) semi-lattice, and whether a lattice has distributive, modular, and Boolean properties. Finally (in § 4) we give Algol realizations of the various algorithms.

Keywords

Computational Mathematic Matrix Representation Computational Analysis Chain Structure Maximal Chain 
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.

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References

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    G. Birkhoff,Lattice Theory, Amer. Math. Soc. Colloq. Publ. XXV, rev. ed. 1961.Google Scholar
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    E. Szpilrajn,Sur l'extension de l'ordre partiel, Fund. Math. 16 (1930), 386–389.Google Scholar
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    R. M. Baer,Certain homomorphisms onto chains, Arch. Math. 8 (1957), 93–95.CrossRefGoogle Scholar
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    A Proposal for Input-Output conventions in ALGOL 60, Comm. ACM, 7 (1964), 273–283.Google Scholar
  5. 5.
    Control Data Corp., 3000/6000 Algol Generic Reference Manual, Dec. 1967, Pub. No. 60214900.Google Scholar

Copyright information

© BIT Foundations 1969

Authors and Affiliations

  • R. M. Baer
    • 1
    • 2
  • O. Østerby
    • 1
    • 2
  1. 1.Computer CenterUniversity of CaliforniaBerkeleyUSA
  2. 2.Matematisk InstitutÅrhus UniversitetÅrhusDenmark

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