Reduced product

Remarks

The relation $\theta_F$ is a congruence on $\mathbf{P}$. Reflexivity uses $I \in F$, and transitivity uses that $F$ is closed under finite intersections and supersets. Compatibility holds because $\set{i : f(a_1, \dots, a_n)(i) = f(b_1, \dots, b_n)(i)}$ contains the intersection of the sets $\set{i : a_k(i) = b_k(i)}$.

Example: if $F = \set{I}$, then $\theta_F$ is equality and the reduced product is the direct product itself. If $F = \set{X \subseteq I : J \subseteq X}$ for some nonempty $J \subseteq I$, the reduced product is isomorphic to $\prod_{j \in J} \mathbf{A}_j$.

Non-example: on $I = \set{1, 2, 3}$, the family $G = \set{\set{1, 2}, \set{2, 3}, I}$ is not a filter, since $\set{1, 2} \cap \set{2, 3} = \set{2} \notin G$. With it, the analogous relation on $\Z^3$ is not transitive: $(0, 0, 0)$ is related to $(0, 0, 1)$, which is related to $(1, 0, 1)$, but $(0, 0, 0)$ and $(1, 0, 1)$ agree only on $\set{2}$.

When $F$ is an ultrafilter, the reduced product is the ultraproduct $\prod_{i \in I} \mathbf{A}_i / F$.