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 {x_P : PP}, and the set of relations is {x_Px_q=x_P : 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.

Keywords: poset algebras; superatomic Boolean algebras; scattered posets; well quasi orderings

Document Type: Research article

DOI: 10.1023/B:ORDE.0000026462.71837.18

Affiliations: 1: Department of Mathematics, Ben Gurion University, Beer-Sheva, Israel., Email: abraham@math.bgu.ac.il 2: Laboratoire de Mathématiques, Université de Savoie, Le Bourget-du-Lac, France., Email: bonnet@in2p3.fr 3: Department of Mathematics, Ben Gurion University, Beer-Sheva, Israel., Email: kubis@math.bgu.ac.il 4: Department of Mathematics, Ben Gurion University, Beer-Sheva, Israel., Email: matti@math.bgu.ac.il