Pointwise order ($f⪯g⟺\forall x,f(x)⪯g(x)$)

Given a partially ordered set $B$ and a set $A$, the pointwise order on the function space of functions $A\to B$ compares their values at every argument. It is a partial order: reflexivity and transitivity hold at every argument, and antisymmetry follows because two functions equal at every argument are equal. Even if $B$ has a total order, this function space need not: the two functions on $\{0,1\}$ with values $(0,1)$ and $(1,0)$ are incomparable when $B=\mathbb{R}$.