# Commutative B-Algebra Induces Abelian Group

## Theorem

Let $\left({X, \circ }\right)$ be a commutative $B$-algebra.

Let $*$ be the binary operation on $X$ defined as:

$\forall a, b \in X: a * b := a \circ \left({0 \circ b}\right)$

Then the algebraic structure $\left({X, *}\right)$ is an abelian group such that:

$\forall x \in X: 0 \circ x$ is the inverse element of $x$ under $*$.

That is:

$\forall a, b \in X: a * b^{-1} := a \circ b$

## Proof

From B-Algebra Induces Group, the algebraic structure $\left({X, *}\right)$ is a group such that:

$\forall x \in X: 0 \circ x$ is the inverse element of $x$ under $*$.

It remains to show that $*$ is a commutative operation.

Let $x, y \in X$:

 $\displaystyle x * y$ $=$ $\displaystyle x \circ \left({0 \circ y}\right)$ by definition of $*$ $\displaystyle$ $=$ $\displaystyle y \circ \left({0 \circ x}\right)$ by definition of commutative $B$-algebra $\displaystyle$ $=$ $\displaystyle y * x$ by definition of $*$

Hence the result.

$\blacksquare$