## 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.

## Preview

Unable to display preview. Download preview PDF.

### References

- [AB1]Abraham, U. and Bonnet, R.: Every superatomic subalgebra of an interval algebra is embeddable in an ordinal algebra,
*Proc. Amer. Math. Soc.***115**(3) (1992), 585–592.MATHMathSciNetCrossRefGoogle Scholar - [AB2]Abraham, U. and Bonnet, R.: A generalization of Hausdorff theorem on scattered poset,
*Fund. Math.***159**(1999), 51–69.MATHMathSciNetGoogle Scholar - [ABK]Abraham, U., Bonnet, R. and Kubiś, W.: Poset algebras over well quasi-ordered posets, Preprint.Google Scholar
- [BR]Bonnet, R. and Rubin, M.: On well-generated Boolean algebras,
*Ann. Pure Appl. Logic***105**(2000), 1–50.MATHMathSciNetCrossRefGoogle Scholar - [F]Fraïssé, R.:
*Theory of Relations*, rev. Edn, with an appendix by Norbert Sauer, Stud. Logic Found. Math., North-Holland, Amsterdam, 2000, ii + 451 pp.Google Scholar - [H]Higman, G.: Ordering by divisibility in abstract algebras,
*Proc. London Math. Soc.***2**(1952), 326–336.MATHMathSciNetGoogle Scholar - [Ko]Koppelberg, S.: In: J. D. Monk (ed.),
*Handbook on Boolean Algebras*, Vol. 1, North-Holland, Amsterdam, 1989.Google Scholar - [P]Pouzet,.: On the set of initial intervals of a scattered poset satisfying FAC, 1981, Private communication. Announced in I. Rival (ed.),
*Ordered Sets (Banff, Alta, 1981)*, NATOAdv. Sci. Inst., Ser. C Math. Phys. 83, D. Reidel, Dordrecht, 1982, p. 847.Google Scholar