Structure of summable tall ideals under kattov order

We show that Katetov and Rudin-Blass orders on summable tall ideals coincide. We prove that Katetov order on summable tall ideals is Galois-Tukey equivalent to $(\omega <^>\omega ,\le <^>*)$. It follows that Katetov 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 Katetov order on summable tall ideals.