B-Algebra Induced by Group Induced by B-Algebra

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\left({S, *}\right)$ be a $B$-algebra.

Let $\left({S, \circ}\right)$ be the group described on $B$-Algebra Induces Group.

Let $\left({S, *'}\right)$ be the $B$-algebra described on Group Induces $B$-Algebra.


Then $\left({S, *'}\right) = \left({S, *}\right)$.


Proof

Let $a, b \in S$.

It is required to show that:

$a *' b = a * b$


To achieve this, recall that $*'$ is defined on Group Induces $B$-Algebra to satisfy:

$a *' b = a \circ b^{-1}$

which, by the definition of $\circ$ on $B$-Algebra Induces Group comes down to:

\(\displaystyle a *' b\) \(=\) \(\displaystyle a * \left({0 * b^{-1} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle a * \left({0 * \left({0 * b}\right)}\right)\) $B$-Algebra Induces Group
\(\displaystyle \) \(=\) \(\displaystyle a * b\) Identity: $0 * \left({0 * x}\right) = x$


Hence the result.

$\blacksquare$


Also see