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

Abstract

We consider the Katětov order between ideals of subsets of natural numbers ("≤K ") and its stronger variant-containing an isomorphic ideal (" "). In particular, we are interested in ideals I for which I ≤K J ⇒ I J for every ideal 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 " " (and vice versa).

Ideals on countable sets: a survey with questions

“…We note that the equivalence of (i) and (ii) was proven in [43] (see section 5.1 in [31]), and that (ii) is equivalent to (iii) for analytic P -ideals was proven in [19,Theorem 4.2]. But the result was formally stated in [1,Proposition 6.5]). This motivates a reformulation of Question 7.8 as follows (see also Theorem 8.3).…”

Homogeneous ideals on countable sets

“…Let us recall that if J 1 is a dense ideal, then J 1 J 2 if and only if there is a 1 − 1 function f : [2] or [3]). Therefore, by the above fact we get that I|A I|B.…”

On a conjecture of Debs and Saint Raymond

“…where A (x) = {y ∈ J : (x, y) ∈ A}. Isomorphisms of ideals have been deeply studied for instance in [11] (see also [4]), while the preorder ⊑ is examined in [1], [2] and [12].…”