Definition 7.1.5 (Product Ideal).label Let $X, Y$ be sets, $\sigma \subset 2^{X}$ and $\tau \subset 2^{Y}$ be ideals, and
\[\beta = \bracs{A \times B|A \in \sigma, B \in \tau}\]
then there exists a unique ideal $\sigma \times \tau$ such that $\beta$ is fundamental with respect to $\sigma$. The ideal $\sigma \otimes \tau$ is the product of $\sigma$ and $\tau$.
Proof. For each $A_{1}, A_{2} \in \sigma$ and $B_{1}, B_{2} \in \tau$,
\[(A_{1} \times B_{1}) \cup (A_{2} \times B_{2}) \subset (A_{1} \cup A_{2}) \times (B_{1} \cup B_{2})\]
By Proposition 7.1.4, there exists an ideal $\sigma \otimes \tau$ such that $\beta$ is fundamental with respect to $\sigma \otimes \tau$.$\square$
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