In an earlier thread, I asked if there was a standard generalization of the absolute value of $\mathbb{C}$ that could be placed on a field, but might not take values in $R_{\geq 0}$. What somebody suggested was that I look into the theory of valuations, and so I went to the Wikipedia page. The definition it gives is as follows.
Let $(G, \cdot, \geq)$ be an abelian, totally ordered group. Define the ordered monoid $G \cup \{ O \}$ by \begin{align*} (\forall g \in G) & (g \geq O) , \\ (\forall g \in G) & (Og = gO = O) . \end{align*} Now let $k$ be a field. Then a map $v : k \to G \cup \{ O \}$ is a valuation on $k$ if \begin{align*} v(a) = O & \iff a = 0 , \\ (\forall a, b \in k) & v(ab) = v(a)v(b) , \\ (\forall a, b \in k) & v(a + b) \leq \max \{ v(a) , v(b) \}. \end{align*}
My issue is that I understood these valuations to be a generalization of the standard idea of putting an absolute value on $\mathbb{C}$ (and its subfields). However, the absolute value would fail the last of these axioms, i.e. $| 1 + 1 | = 2 \not \leq \max \{ |1| , |1| \}$. Am I misreading this? Or is this whole valuation thing just completely different than absolute value?