## 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 } : *p*∈*P*}, and the set of relations is {*x* _{ p }⋅*x* _{ q }=*x* _{ p } : *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*,≤^{ 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.

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