Number of Ultrafilters

Abstract

This cardinal function is rather easy to describe, at least if we do not try to go into the detail that we did for cellularity, for example. If A is a subalgebra or homomorphic image of B, then |UltA| ≤ |UltB|. For weak products we have

$$|\Pi _{i \in I}^w{A_i}| = \max (\omega ,{\sup _{i \in I}}|Ult{A_i}|)$$

. The situation for full products is more complicated:

$$|Ult(\mathop \Pi \limits_{i \in I} {A_i})| \leqslant {2^{{2^k}}}$$

, where κ = Σ_i∈I dA _i . This follows from the following two facts: where “→” means “is isomorphically embeddable in”, and “U” means “disjoint union”. Next, clearly |Ult ⊕_i∈I = Π_i∈I |UltA _i |.