PhyslibAlpha.AlgebraicFramework.OrderUnit.Channel.Normal

Let $E_1$ and $E_2$ be ordered vector spaces. A unital positive linear map $\phi: E_1 \to E_2$ is said to be **normal** if it preserves the suprema of directed sets. Specifically, for every non-empty directed subset $D \subseteq E_1$, if $x$ is the least upper bound of $D$ ($x = \sup D$), then $\phi(x)$ is the least upper bound of the image set $\phi(D)$ ($\phi(x) = \sup \phi(D)$).

Let $E_1$ be an ordered vector space. The identity unital positive linear map $\text{id} : E_1 \to E_1$ is normal. That is, for every non-empty directed subset $D \subseteq E_1$, if $x$ is the least upper bound of $D$ ($x = \sup D$), then the identity map preserves this supremum such that $\text{id}(x) = \sup(\text{id}(D))$.

Let $E_1, E_2,$ and $E_3$ be ordered vector spaces. Suppose $\phi: E_1 \to E_2$ and $\psi: E_2 \to E_3$ are unital positive linear maps. If both $\phi$ and $\psi$ are normal (meaning they preserve the suprema of non-empty directed sets), then their composition $\psi \circ \phi: E_1 \to E_3$ is also normal.