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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References
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.
Abraham, U. and Bonnet, R.: A generalization of Hausdorff theorem on scattered poset, Fund. Math. 159 (1999), 51–69.
Abraham, U., Bonnet, R. and Kubiś, W.: Poset algebras over well quasi-ordered posets, Preprint.
Bonnet, R. and Rubin, M.: On well-generated Boolean algebras, Ann. Pure Appl. Logic 105 (2000), 1–50.
Fraïssé, R.: Theory of Relations, rev. Edn, with an appendix by Norbert Sauer, Stud. Logic Found. Math., North-Holland, Amsterdam, 2000, ii + 451 pp.
Higman, G.: Ordering by divisibility in abstract algebras, Proc. London Math. Soc. 2 (1952), 326–336.
Koppelberg, S.: In: J. D. Monk (ed.), Handbook on Boolean Algebras, Vol. 1, North-Holland, Amsterdam, 1989.
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.
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Abraham, U., Bonnet, R., Kubiś, W. et al. On Poset Boolean Algebras. Order 20, 265–290 (2003). https://doi.org/10.1023/B:ORDE.0000026462.71837.18
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DOI: https://doi.org/10.1023/B:ORDE.0000026462.71837.18