## 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}Upwards harpoon with barb rightwardsG〉 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.

Original language | English |
---|---|

Pages (from-to) | 265-290 |

Number of pages | 26 |

Journal | Order |

Volume | 20 |

Issue number | 3 |

DOIs | |

State | Published - 1 Dec 2003 |

## Keywords

- Poset algebras
- Scattered posets
- Superatomic Boolean algebras
- Well quasi orderings

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics