Banach lattice

Remarks

Clauses 1 to 3 say that $E$ is a Riesz space, and clause 4 says that the norm is a lattice norm.

Example: $C(K)$ for a compact Hausdorff space $K$, with $f \le g$ when $f(t) \le g(t)$ for all $t \in K$. Clauses 1 and 2 are clear, and $f \vee g$ is the pointwise maximum, which is continuous. Then $\abs{f}$ is the pointwise absolute value, and $\abs{f} \le \abs{g}$ gives $\sup \abs{f} \le \sup \abs{g}$.

Non-example: $\R^2$ with the coordinatewise order and the norm $\norm{(a, b)} = \abs{a} + \abs{a - b}$. Clauses 1 to 3 hold, and this is a norm. For $x = (1, 0)$ and $y = (1, 1)$, one has $\abs{x} \le \abs{y}$, but $\norm{x} = 2 > 1 = \norm{y}$, so clause 4 fails.