When does the Katětov order imply that one ideal extends the other?

Paweł Barbarski ¹   ; Rafał Filipów ¹   ; Nikodem Mrożek ¹   ; Piotr Szuca ¹

¹ Institute of Mathematics University of Gdańsk Wita Stwosza 57 80-952 Gdańsk, Poland

Colloquium Mathematicum, Tome 130 (2013) no. 1, pp. 91-102

Résumé

We consider the Katětov order between ideals of subsets of natural numbers (“$\leq_{K}$”) and its stronger variant—containing an isomorphic ideal (“$\sqsubseteq$”). In particular, we are interested in ideals $\mathcal{I}$ for which $$ \mathcal{I}\leq_{K}\mathcal{J}\ \Rightarrow\ \mathcal{I}\sqsubseteq\mathcal{J} $$ for every ideal $\mathcal{J}$. We find examples of ideals with this property and show how this property can be used to reformulate some problems known from the literature in terms of the Katětov order instead of the order “$\sqsubseteq$” (and vice versa).

Détail

DOI : 10.4064/cm130-1-9

Keywords: consider kat tov order between ideals subsets natural numbers leq its stronger variant containing isomorphic ideal sqsubseteq particular interested ideals mathcal which mathcal leq mathcal rightarrow mathcal sqsubseteq mathcal every ideal mathcal examples ideals property property reformulate problems known literature terms kat tov order instead order sqsubseteq vice versa