Definition:Set Product

Definition
Let $S$ and $T$ be sets.

Let $P$ be a set and let $\phi_1: P \to S$ and $\phi_2: P \to T$ be mappings such that:


 * For all sets $X$ and all mappings $f_1: X \to S$ and $f_2: X \to T$ there exists a unique mapping $h: X \to P$ such that:
 * $\phi_1 \circ h = f_1$
 * $\phi_2 \circ h = f_2$


 * that is, such that:


 * $\begin{xy}\xymatrix@+1em@L+3px{

& X \ar[ld]_*+{f_1} \ar@{-->}[d]^*+{h} \ar[rd]^*+{f_2}

\\ S & P \ar[l]^*+{\phi_1} \ar[r]_*+{\phi_2} & T }\end{xy}$


 * is a commutative diagram.

Then $P$, together with the mappings $\phi_1$ and $\phi_2$, is called a product of $S$ and $T$.

This product of $S$ and $T$ can be denoted $\struct {P, \phi_1, \phi_2}$.

Also see

 * Definition:Cartesian Product
 * Definition:Product (Category Theory), a generalization to any metacategory.