On Poset Boolean Algebras

Let 〈P,≤〉 be a partially ordered set. We define the poset Boolean algebra of P , and denote it by F (P ). The set of generators of F (P ) is {xp : p ∈ P}, and the set of relations is {xp ·xq = xp : p ≤ q}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and 〈G,≤G〉 is well-founded. A wellgenerated 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 [P]. A partially ordered set P is well-ordered, if P does not contain a strictly decreasing infinite sequence, and P 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 partly analogous to the fact (proved in [AB1]), that a superatomic interval algebra is isomorphic to a subalgebra of an ordinal algebra. Mathematics Subject Classification 2000 (MSC2000) Primary: 03G05, 06A06. Secondary: 06E05, 08A05, 54G12.