Some Applications of the Katětov Order on Borel Ideals

We construct an embedding of the algebra $\mathcal {P}(\omega )/\mathrm {Fin}$ into the family of summable ideals with the Katětov order. This construction will be used to solve two problems: about the relation between the Katětov order and the ideal Baire classes of functions, and about long chains of ideals alternately with and without the property of being a P-ideal.