Physlib.ProbabilisticTheory.OrderUnit.Cone

Given an ordered real vector space $E$, the positive cone $\text{PosCone}(E)$ is defined as the set of all non-negative elements $\{x \in E \mid 0 \le x\}$. This set forms a pointed cone over the real numbers $\mathbb{R}$, meaning it contains the zero element and is closed under addition and scalar multiplication by non-negative real numbers.

Given an ordered real vector space $E$, the positive cone $\text{PosCone}(E) = \{x \in E \mid 0 \le x\}$ possesses the structure of a module over the semiring of non-negative real numbers $\mathbb{R}_{\ge 0}$. This means that the cone is closed under addition and carries a scalar action such that for any $c \in \mathbb{R}_{\ge 0}$ and $x \in \text{PosCone}(E)$, the element $c \cdot x$ remains within the positive cone.

In an Archimedean order-unit space $E$, the positive cone $\text{PosCone}(E)$, defined as the set of elements $\{x \in E \mid 0 \le x\}$, is a closed set with respect to the topology induced by the order-unit norm on $E$.

The identity element (order unit) $1$ of an ordered space $E$ is defined as an element of the positive cone $\text{PosCone}(E)$. This construction represents the order unit $1$ as a member of the set of non-negative elements $\{x \in E \mid 0 \le x\}$.

In a space $E$ equipped with a unit element $1$, the unit element $1$ of the positive cone $\text{PosCone}(E)$, when viewed as an element of $E$ via the canonical coercion, is equal to the unit element $1$ of $E$.