# Category:B-Algebras

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This category contains results about **B-Algebras**.

Definitions specific to this category can be found in Definitions/B-Algebras.

Let $\struct {X, \circ}$ be an algebraic structure.

$\struct {X, \circ}$ is a **$B$-algebra** if and only if $\struct {X, \circ}$ satisfies the $B$-algebra axioms:

\((\text {AC})\) | $:$ | \(\ds \forall x, y \in X:\) | \(\ds x \circ y \in X \) | ||||||

\((\text A 0)\) | $:$ | \(\ds \exists 0 \in X \) | |||||||

\((\text A 1)\) | $:$ | \(\ds \forall x \in X:\) | \(\ds x \circ x = 0 \) | ||||||

\((\text A 2)\) | $:$ | \(\ds \forall x \in X:\) | \(\ds x \circ 0 = x \) | ||||||

\((\text A 3)\) | $:$ | \(\ds \forall x, y, z \in X:\) | \(\ds \paren {x \circ y} \circ z = x \circ \paren {z \circ \paren {0 \circ y} } \) |

## Pages in category "B-Algebras"

The following 26 pages are in this category, out of 26 total.

### B

- B-Algebra Identity: 0(0x)=x
- B-Algebra Identity: x (y z) = (x (0 z)) y
- B-Algebra Identity: xy = 0 iff x = y
- B-Algebra Identity: xy=x(0(0y))
- B-Algebra Induced by Group Induced by B-Algebra
- B-Algebra Induces Group
- B-Algebra is Commutative iff x(xy)=y
- B-Algebra is Left Cancellable
- B-Algebra is Quasigroup
- B-Algebra is Right Cancellable
- B-Algebra Power Law
- B-Algebra Power Law with Zero
- B-Algebra/Examples
- B-Algebra/Examples/S3