Cardinality of reduced products

Question

Suppose $\mathcal{A}_i (i \in I)$ is a family of $L$-structures and consider the reduced product $\mathcal{A}$ of this family by a filter $F \subseteq \mathcal{P}(I)$. Is there a way to determine the cardinality of $\mathcal{A}$ in terms of the cardinality of each of its factors? For example, suppose each $\mathcal{A}_i$ is finite, but there is no bound on their cardinalities. Does it follow that $\mathcal{A}$ is infinite?

Answer 1

For example, suppose each $\mathcal{A}_i$ is finite, but there is no bound on their cardinalities. Does it follow that $\mathcal{A}$ is infinite?

No, not necessarily. For example, consider $\mathcal A_i = \{0, 1, \ldots, i \}$ and let $F \subseteq \mathcal{P}(\mathbb{N})$ be the principal ultrafilter given by $$ x \in F \iff 0 \in x. $$ Then, for any $[f]$ in the reduced ultrapower, we have $$ \{ n \in \mathbb N \mid f(n) = 0 \} \ni 0 $$ and hence $[f] = [\{(0,0)\}]$, i.e. $\prod_{i \in \mathbb N} \mathcal{A}_i / F = \{[(0,0)] \}$ has cardinality $1$.

On the other hand, clearly $\prod_{i \in I} \mathcal{A}_i / F$ has cardinality $\le \mathrm{card}(\prod_{i \in I} \mathcal{A}_i)$ and equality is possible.