Let \$$\mathcal{P} := \langle P, \preceq_P \rangle\$$ and \$$\mathcal{Q} := \langle Q, \preceq_Q \rangle\$$. Define \$$\mathcal{P} \times \mathcal{Q} := \langle P \times Q, \preceq_{P\times Q} \rangle\$$ where:

\$(a,b) \preceq_{P\times Q} (c,d) \text{ if and only if } a \preceq_P c \text{ and } b \preceq_Q d \$

We need to check reflexivity and transitivity.

_Reflexivity_ - Since \$$a \preceq_P a\$$ and \$$b \preceq_Q b\$$ then \$$(a,b) \preceq_{P \times Q} (a,b)\ \ \ \Box\$$

_Transitivity_ - Let \$$(a,b) \preceq_{P \times Q} (c,d)\$$ and \$$(c,d) \preceq_{P \times Q} (e,f)\$$. Then we have \$$a \preceq_P c \preceq_P e\$$ and \$$b \preceq_Q d \preceq_Q f\$$.

Since \$$\preceq_P\$$ and \$$\preceq_Q\$$ are transitive, then \$$a \preceq_P e\$$ and \$$b \preceq_P f\$$.

Hence \$$(a,b) \preceq_{P \times Q} (e,f)\ \ \ \Box\$$

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We can also show that this product order is the product in the category of preorders where the morphisms are order-preserving maps.

In fact, the category of preorders is _bicartesian closed_. There's an initial preorder, a final preorder, products and coproducts and an exponential. If this is interesting we can go into this construction and the proof.