Cellularity and the structure of pseudo-trees

Abstract

Let T be an infinite pseudo-tree. In [2], we showed that the cellularity of the pseudo-tree algebra Treealg(T) was the maximum of four cardinals c_T, ι_T, φ_T, and μ_T: roughly, c_T is the “tallness” of T; ι_T is the “width” of T; φ is the number of “points of finite branching” in T; and μ is the number of “sections of no branching” in T. Here we ask: which inequalities among these four cardinals may be satisfied, in some sense, by a pseudo-tree? We show that the possible inequalities among c_T, ι_T, φ_T, and μ_T attained by pseudo-trees T are closely related to the existence of generalized Suslin trees.