@article {Abraham:2003:0167-8094:265, author = "Abraham U.", author = "Bonnet R.", author = "Kubis W.", author = "Rubin M.", title = "On Poset Boolean Algebras", journal = "Order", volume = "20", year = "2003", abstract = "Let langP,lesrang 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 : PisinP}, and the set of relations is {xPsdotxq=xP : Plesq}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and langG,lesB|Grang is well-founded. A well-generated algebra is superatomic.

THEOREM 1. Let langP,lesrang 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) hArr (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.", pages = "265-290(26)", url = "http://www.ingentaconnect.com/content/klu/orde/2003/00000020/00000003/05139036" doi = "doi:10.1023/B:ORDE.0000026462.71837.18" }