When does the Katětov order imply that one ideal extends the other?
Volume 130 / 2013
Colloquium Mathematicum 130 (2013), 91-102
MSC: Primary 03E05;
Secondary
03E15,
40A35.
DOI: 10.4064/cm130-1-9
Abstract
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).
