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\\)


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.