BF-algebra

In mathematics, BF algebras are a class of algebraic structures arising out of a symmetric "Yin Yang" concept for Bipolar Fuzzy logic, the name was introduced by Andrzej Walendziak in 2007. The name covers discrete versions, but a canonical example arises in the BF space [-1,0]x[0,1] of pairs of (false-ness, truth-ness).

Definition
A BF-algebra is a non-empty subset $$X$$ with a constant $$0$$ and a binary operation $$*$$ satisfying the following:
 * 1) $$x*x=0$$
 * 2) $$x*0=x$$
 * 3) $$0*(x*y)=y*x$$

Example
Let $$Z$$ be the set of integers and '$$-$$' be the binary operation 'subtraction'. Then the algebraic structure $$(Z,-)$$ obeys the following properties:
 * 1) $$x-x=0$$
 * 2) $$x-0=x$$
 * 3) $$0-(x-y)=y-x$$