PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.Hom

Given two unital real Jordan algebras $E$ and $F$, an instance of `CoeFun` allows a Jordan homomorphism $f \in \text{JordanHom}(E, F)$ to be applied as a function from $E$ to $F$.

Let $E$ and $F$ be unital real Jordan algebras. For any two Jordan homomorphisms $f, g: E \to F$, if $f(x) = g(x)$ for all $x \in E$, then $f = g$.

Let $E$ and $F$ be real Jordan algebras. For any Jordan homomorphism $f : E \to F$, the image of the zero element in $E$ is the zero element in $F$, that is, $f(0) = 0$.

Let $E$ and $F$ be unital real Jordan algebras, and let $f: E \to F$ be a Jordan homomorphism. For any $x, y \in E$, the map $f$ preserves addition, such that $f(x + y) = f(x) + f(y)$.

Let $E$ and $F$ be unital real Jordan algebras, and let $f: E \to F$ be a Jordan homomorphism. For any scalar $r \in \mathbb{R}$ and any element $x \in E$, the image of the scalar multiplication $r \cdot x$ under $f$ is equal to $r \cdot f(x)$.

Let $E$ and $F$ be unital real Jordan algebras. For any Jordan homomorphism $f : E \to F$, the image of the identity element $1$ in $E$ is the identity element $1$ in $F$, that is, $f(1) = 1$.

Let $E$ and $F$ be unital real Jordan algebras. For any Jordan homomorphism $f : E \to F$ and any elements $x, y \in E$, the map preserves the Jordan product such that $f(x \cdot y) = f(x) \cdot f(y)$.

For a unital real Jordan algebra $E$, the identity Jordan homomorphism $\text{id}: E \to E$ is the map that sends every element $x \in E$ to itself. This map preserves the identity element ($1$) and the Jordan product.

Given two unital real Jordan homomorphisms $f: E \to F$ and $g: F \to G$ between unital real Jordan algebras $E, F$, and $G$, their composition $g \circ f$ is a unital real Jordan homomorphism from $E$ to $G$.

For a unital real Jordan algebra $E$ and any element $x \in E$, the identity Jordan homomorphism $\text{id} : E \to E$ applied to $x$ yields $x$, that is, $\text{id}(x) = x$.

Let $E, F,$ and $G$ be unital real Jordan algebras. For any Jordan homomorphisms $f: E \to F$ and $g: F \to G$ and any element $x \in E$, the value of the composed homomorphism $g \circ f$ at $x$ is equal to $g(f(x))$.

Let $E$ and $F$ be unital real Jordan algebras. For any Jordan homomorphism $f : E \to F$, the composition of the identity Jordan homomorphism $\text{id}_F$ on $F$ with $f$ is equal to $f$. In mathematical notation, this is expressed as $\text{id}_F \circ f = f$.

Let $E$ and $F$ be unital real Jordan algebras. For any Jordan homomorphism $f: E \to F$, its composition with the identity Jordan homomorphism $\text{id}_E: E \to E$ results in the original homomorphism $f$, that is, $f \circ \text{id}_E = f$.

Let $E, F$, and $G$ be unital real Jordan algebras. For any Jordan homomorphisms (unital real-linear maps that preserve the Jordan product) $f: E \to F$, $g: F \to G$, and $h: G \to E$, the composition of these maps is associative: $$(h \circ g) \circ f = h \circ (g \circ f)$$