PhyslibAlpha.ProbabilisticTheory.JordanOrderUnit.Power.Ring

Given an element $a$ in a Jordan algebra $E$ over $\mathbb{R}$, the type `GeneratedByOne a` (denoted as $J[a]$) is defined as the subtype of $E$ consisting of elements in the $\mathbb{R}$-submodule spanned by the set of Jordan powers $\{a^n\}_{n \in \mathbb{N}}$, where $a^0 = 1$ and $a^{n+1} = a \circ a^n$. While the ambient algebra $E$ is generally non-associative, $J[a]$ is a commutative and associative unital $\mathbb{R}$-algebra.

Given an element $a$ in a Jordan algebra $E$, this definition provides the multiplication operation on the commutative associative subalgebra $J[a]$ (the $\mathbb{R}$-submodule spanned by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$). For elements $x, y \in J[a]$, their product is defined by the Jordan product $x \circ y$ in the ambient algebra $E$.

Let $E$ be a Jordan algebra over $\mathbb{R}$ and $a \in E$. Let $J[a]$ denote the commutative associative subalgebra of $E$ generated by the powers of $a$. For any elements $x, y \in J[a]$, the product $x \cdot y$ as defined in $J[a]$ is equal to the Jordan product $x \circ y$ in the ambient algebra $E$.

This definition provides the multiplicative identity element $1$ for the subalgebra $J[a]$ (represented by the type `GeneratedByOne a`), which is the $\mathbb{R}$-submodule of a Jordan algebra $E$ spanned by the powers of an element $a$. The identity of the subalgebra is inherited from the identity element of the ambient algebra $E$.

In a Jordan algebra $E$ over $\mathbb{R}$, let $J[a]$ (represented by the type `GeneratedByOne a`) be the commutative associative subalgebra spanned by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$ of an element $a \in E$. The multiplicative identity element $1$ of the subalgebra $J[a]$ is equal to the multiplicative identity element $1$ of the ambient Jordan algebra $E$.

Let $E$ be a Jordan algebra over $\mathbb{R}$ and $a \in E$ be an element. Let $J[a]$ denote the subalgebra generated by $a$ (defined as the $\mathbb{R}$-submodule spanned by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$). For any elements $x, y, z \in J[a]$, the multiplication operation satisfies the associative law: $$(x \cdot y) \cdot z = x \cdot (y \cdot z)$$

Let $E$ be a Jordan algebra over $\mathbb{R}$ and let $a$ be an element in $E$. Let $J[a]$ denote the subalgebra spanned by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$ of $a$. For any elements $x, y \in J[a]$, the multiplication operation is commutative, satisfying $x \cdot y = y \cdot x$.

Let $a$ be an element of a Jordan algebra $E$ and let $J[a]$ denote the commutative associative subalgebra generated by $a$. For any element $x \in J[a]$, the multiplicative identity $1$ of the subalgebra satisfies $1 \cdot x = x$.

Let $a$ be an element of a Jordan algebra $E$, and let $J[a]$ denote the commutative associative unital $\mathbb{R}$-subalgebra generated by $a$. For any element $x \in J[a]$, it holds that $x \cdot 1 = x$, where $1$ is the identity element of $J[a]$.

Let $J[a]$ be the commutative associative subalgebra of a Jordan algebra $E$ over $\mathbb{R}$ spanned by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$ of an element $a \in E$. For any elements $x, y, z \in J[a]$, the multiplication operation (inherited from the Jordan product) distributes over addition from the left: $$x \cdot (y + z) = x \cdot y + x \cdot z$$

Let $J[a]$ be the commutative associative subalgebra of a Jordan algebra $E$ over $\mathbb{R}$ spanned by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$ of an element $a \in E$. For any elements $x, y, z \in J[a]$, the multiplication operation (inherited from the Jordan product) distributes over addition from the right: $$(x + y) \cdot z = x \cdot z + y \cdot z$$

Given a Jordan algebra $E$ over $\mathbb{R}$ and an element $a \in E$, the subtype $J[a]$ (represented by `GeneratedByOne a`), which is the $\mathbb{R}$-submodule of $E$ spanned by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$, forms a commutative ring. The multiplication in this ring is defined by the Jordan product $x \circ y$, which is associative when restricted to $J[a]$, and the identity element is the same as the identity of the ambient algebra $E$.

Given an element $a$ in a Jordan algebra $E$ over $\mathbb{R}$, let $J[a]$ be the commutative associative unital $\mathbb{R}$-subalgebra generated by $a$ and the identity. This definition represents the natural inclusion map $J[a] \hookrightarrow E$, which maps each element $x \in J[a]$ to its corresponding value in the ambient algebra $E$.

Let $E$ be a Jordan algebra over $\mathbb{R}$ and $a \in E$. Let $J[a]$ be the commutative associative unital $\mathbb{R}$-subalgebra generated by $a$ and the identity. For any element $x \in J[a]$, the inclusion map $\iota : J[a] \to E$ satisfies $\iota(x) = x$, where the right-hand side represents the element $x$ viewed as a member of the ambient algebra $E$.

Let $E$ be a Jordan algebra over $\mathbb{R}$ and $a \in E$. Let $J[a]$ be the commutative associative unital $\mathbb{R}$-subalgebra generated by the powers of $a$. Then the inclusion map $\iota : J[a] \to E$ is injective.

Let $E$ be a Jordan algebra over $\mathbb{R}$ and $a \in E$. The subtype $J[a]$ (represented by `GeneratedByOne a`), which is the $\mathbb{R}$-submodule of $E$ spanned by the Jordan powers $\{a^n\}_{n \in \mathbb{N}}$, is equipped with an $\mathbb{R}$-algebra structure. The algebra map $\eta: \mathbb{R} \to J[a]$ is defined by $\eta(c) = c \cdot 1$, where $1$ is the identity element of $E$. The scalar multiplication of $c \in \mathbb{R}$ on $x \in J[a]$ is inherited from the ambient module structure of $E$ and satisfies the identity $c \cdot x = (c \cdot 1) \circ x$, where $\circ$ denotes the Jordan product.