# Anti-Chains of Mappings from omega^omega on some BQO

## Details

Serval ID

serval:BIB_FA2CB02AE2CF

Type

**Unpublished**: a document having an author and title, but not formally published.

Collection

Publications

Institution

Title

Anti-Chains of Mappings from omega^omega on some BQO

Language

english

Notes

temporary version, to be submitted

Abstract

Issue: Let (P,<) be some BQO. Louveau and Saint-Raymond showed that the following structure (F, <_F) is also a BQO: F={φ: from ωω into P: φ is Borel with countable image} with the usual topology on ωω and the discrete topology on the BQO P; and φ <_F ψ iff there exists some continuous function h: from ωω to ωω such that for all x ∈ ωω φ(x) < ψ(h(x)). The following proposition answers the question of the relation between cardinalities of anti-chains of P and anti-chains of F: 1) Every anti-chain in P has cardinality 1 ⇒ every anti-chain in F has cardinality 1. 2) There exists an anti-chain in P of cardinality 2, but no element of P is incomparable with two different elements ⇒ every anti-chain in F has cardinality at most 2. 3) There exists an element in P which is incomparable with two different elements ⇒ there exists anti-chains of any cardinality in F.

Create date

23/01/2008 20:40

Last modification date

20/08/2019 17:25