Order Theory/Lexicographic order

Definition (lexicographic order):

Let $(S_{\alpha },{\le }_{\alpha }{)}_{\alpha \in A}$ be preordered sets, where $A$ is well-ordered. Define an order on $\bigcup _{\alpha \in A}$, the Cartesian product, by

$(s_{\alpha }{)}_{\alpha \in A}\le (t_{\alpha }{)}_{\alpha \in A}:\iff$.

Proposition (lexicographic order induced by posets is poset):

Whenever ${\displaystyle }$

Proposition (lexicographic order induced by total orders is total):

Whenever $A$ is a well-ordered set and $(S_{\alpha },{\le }_{\alpha }{)}_{\alpha \in A}$ are totally ordered sets, the lexicographic order on $\prod _{\alpha \in A}{S}_{\alpha }$ is total.

Proof: Let any two elements $(s_{\alpha }{)}_{\alpha \in A}$ and $(t_{\alpha }{)}_{\alpha \in A}$ of $\prod _{\alpha \in A}{S}_{\alpha }$ be given. Then either $(s_{\alpha }{)}_{\alpha \in A}=(t_{\alpha }{)}_{\alpha \in A}$, or there exists a smallest $\beta \in A$ so that $s_{\beta }\ne t_{\beta }$. Since ${\le }_{\beta }$ is total, either $s_{\beta }<t_{\beta }$ or $s_{\beta }>t_{\beta }$, and thus either $(s_{\alpha }{)}_{\alpha \in A}<(t_{\alpha }{)}_{\alpha \in A}$ or $(s_{\alpha }{)}_{\alpha \in A}>(t_{\alpha }{)}_{\alpha \in A}$. $◻$