Towers, mad families, and unboundedness

We show that Hechler’s forcings for adding a tower and for adding a mad family can be represented as finite support iterations of Mathias forcings with respect to filters and that these filters are B-Canjar for any countably directed unbounded family B of the ground model. In particular, they preserve the unboundedness of any unbounded scale of the ground model. Moreover, we show that b=ω1 in every extension by the above forcing notions.