Definition:Cartesian Product of Relations

Definition
If for each $i \in I$, $\sim_i$ is a relation from $S_i$ to $T_i$,

$\displaystyle S = \prod_{i \in I} S_i$, and

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

then the product of the relations $\sim_i$ is defined as the relation $\sim$ from $S$ to $T$ such that $x\sim y$ iff $x_i \sim_i y_i$ for each $i\in I$.

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