From ProofWiki
Jump to: navigation, search

This category contains results about B-Algebras.
Definitions specific to this category can be found in Definitions/B-Algebras.

Let $\left({X, \circ}\right)$ be an algebraic structure.

Then $\left({X, \circ}\right)$ is a $B$-algebra if and only if:

\((AC)\)   $:$     \(\displaystyle \forall x, y \in X:\) \(\displaystyle x \circ y \in X \)             
\((A0)\)   $:$   \(\displaystyle \exists 0 \in X \)             
\((A1)\)   $:$     \(\displaystyle \forall x \in X:\) \(\displaystyle x \circ x = 0 \)             
\((A2)\)   $:$     \(\displaystyle \forall x \in X:\) \(\displaystyle x \circ 0 = x \)             
\((A3)\)   $:$     \(\displaystyle \forall x,y,z \in X:\) \(\displaystyle \left({x \circ y}\right) \circ z = x \circ \left({z \circ \left({0 \circ y}\right)}\right) \)