Order

, Volume 1, Issue 3, pp 259–263 | Cite as

A cofinal coloring theorem for partially ordered algebras

  • George M. Bergman
  • Irving Kaplansky
Article

Abstract

If P is a directed partially ordered algebra of an appropriate sort-e.g. an upper semilattice-and has no maximal element, then P has two disjoint subalgebras each cofinal in P. In fact, if P has cofinality α then there exists a family of α such disjoint subalgebras. A version of this result is also proved without the directedness assumption, in which the cofinality of P is replaced by an invariant which we call its global cofinality.

AMS (MOS) subject classifications (1980)

Primary: 06A10, 06A12, 06F99 secondary: 04A10, 04A20, 06B05, 06F05 

Key words

Partially ordered algebra disjoint cofinal subalgebras lattice semilattice cofinality (resp. global cofinality) of a partially ordered set 

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References

  1. 1.
    E. C.Milner and M.Pouzet (1982) On the cofinality of partially ordered sets, in I.Rival (ed.), Ordered Sets, D. Reidel, Dordrecht, pp. 279–298.Google Scholar
  2. 2.
    George M. Bergman and Ehud Hrushovski (to appear) Identities of cofinal sublattices.Google Scholar
  3. 3.
    George M. Bergman (in preparation) On the amalgamation basis of the category of compact groups.Google Scholar
  4. 4.
    A. H.Stone (1968) On partitioning ordered sets into, cofinal subsets, Mathematika (London) 15, 217–222.Google Scholar

Copyright information

© D. Reidel Publishing Company 1985

Authors and Affiliations

  • George M. Bergman
    • 1
  • Irving Kaplansky
    • 2
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  2. 2.Mathematical Sciences Research InstituteBerkeleyUSA
  3. 3.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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