Monotone hulls for N cap M

TL;DR

This paper proves the consistency of the non-existence of monotone hulls for the intersection of measure and category ideals using advanced set-theoretic methods, and explores conditions under which monotone Borel hulls can exist.

Contribution

It demonstrates the consistency of non-existence of monotone hulls for M cap N and shows conditions for their existence for M and N separately.

Findings

No monotone hulls for M cap N under certain set-theoretic assumptions

Existence of monotone Borel hulls for M and N without tower generation

Use of decisive creatures and FS iteration methods in proofs

Abstract

Using the method of decisive creatures (math.LO/0601083) we show the consistency of "there is no increasing omega_2 --chain of Borel sets and non(N)=non(M)= omega_2=2^omega". Hence, consistently, there are no monotone hulls for the ideal M cap N . This answers Balcerzak and Filipczak. Next we use FS iteration with partial memory to show that there may be monotone Borel hulls for the ideals M, N even if they are not generated by towers.