Abstract We show that Katětov and Rudin–Blass orders on summable tall ideals coincide. We prove that Katětov order on summable tall ideals is Galois–Tukey equivalent to $(\omega ^\omega ,\le ^*)$ . It follows that Katětov order on summable tall ideals is upwards directed which answers a question of Minami and Sakai. In addition, we prove that ${l_\infty }$ is Borel bireducible to an equivalence relation induced by Katětov order on summable tall ideals.

Abstract We investigate the set (ω) of partitions of the natural numbers ordered by ≤* where A ≤* B if by gluing finitely many blocks of A we can get a partition coarser than B. In particular, we determine the values of a number of cardinals which are naturally associated with the structure ((ω), ≥*), in terms of classical cardinal invariants of the continuum.