# Order Theory/Total orders

Definition (total order):

An order ${\displaystyle \leq }$ on a set ${\displaystyle S}$ is said to be total if and only if for each ${\displaystyle x,y\in S}$, exactly one of the possibilities ${\displaystyle x, ${\displaystyle x=y}$ or ${\displaystyle x>y}$ occurs.

{{proposition|series order induced by total orders is total|Whenever ${\displaystyle A}$ is a totally ordered set

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

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

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