Definition:Cartesian Product of Relations

Definition
Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ and $\left\langle{T_i}\right\rangle_{i \mathop \in I}$ be families of sets indexed by $I$.

For each $i \in I$, let $\mathcal R_i \subseteq S_i \times T_i$ be a relation from $S_i$ to $T_i$.

Let $S$ and $T$ be the Cartesian product of $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ and $\left\langle{T_i}\right\rangle_{i \mathop \in I}$ respectively:
 * $\displaystyle S = \prod_{i \mathop \in I} S_i$


 * $\displaystyle T = \prod_{i \mathop \in I} T_i$

Then the product of the relations $\mathcal R_i$ is defined as the relation $\mathcal R \subseteq S \times T$ such that:
 * $x \mathrel{\mathcal R} y \iff \forall i \in I: x_i \mathrel{\mathcal R_i} y_i$

Also see

 * Product of Transitive Relations is Transitive
 * Product of Reflexive Relations is Reflexive
 * Product of Antisymmetric Relations is Antisymmetric
 * Product of Preorders is Preorder
 * Product of Orders is Order
 * Product of Directed Sets is Directed Set