Order

, Volume 20, Issue 3, pp 265–290 | Cite as

On Poset Boolean Algebras

  • Uri Abraham
  • Robert Bonnet
  • Wiesław Kubiś
  • Matatyahu Rubin
Article

Abstract

Let (P,≤) be a partially ordered set. The poset Boolean algebra of P, denoted F(P), is defined as follows: The set of generators of F(P) is {xp : pP}, and the set of relations is {xpxq=xp : pq}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and (G,≤B|G) is well-founded. A well-generated algebra is superatomic.

THEOREM 1. Let (P,≤) be a partially ordered set. The following are equivalent. (i) P does not contain an infinite set of pairwise incomparable elements, and P does not contain a subset isomorphic to the chain of rational numbers, (ii) F(P) is superatomic, (iii) F(P) is well-generated.

The equivalence (i) ⇔ (ii) is due to M. Pouzet. A partially ordered set W is well-ordered, if W does not contain a strictly decreasing infinite sequence, and W does not contain an infinite set of pairwise incomparable elements.

THEOREM 2. Let F(P) be a superatomic poset algebra. Then there are a well-ordered set W and a subalgebra B of F(W), such that F(P) is a homomorphic image of B.

This is similar but weaker than the fact that every interval algebra of a scattered chain is embeddable in an ordinal algebra. Remember that an interval algebra is a special case of a poset algebra.

poset algebras superatomic Boolean algebras scattered posets well quasi orderings 

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References

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Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Uri Abraham
    • 1
  • Robert Bonnet
    • 2
  • Wiesław Kubiś
    • 1
  • Matatyahu Rubin
    • 1
  1. 1.Department of MathematicsBen Gurion UniversityBeer-ShevaIsrael
  2. 2.Laboratoire de MathématiquesUniversité de SavoieLe Bourget-du-LacFrance

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