BF-algebra - Wikipedia

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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

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A BF-algebra is a non-empty subset $X$ with a constant $0$ and a binary operation $*$ satisfying the following:

$x*x=0$
$x*0=x$
$0*(x*y)=y*x$

Example

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Let $Z$ be the set of integers and ' $-$ ' be the binary operation 'subtraction'. Then the algebraic structure $(Z,-)$ obeys the following properties:

$x-x=0$
$x-0=x$
$0-(x-y)=y-x$

References

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Walendziak, Andrzej (2007), "On BF-algebras", Math. Slovaca, 57 (2): 119–128, doi:10.2478/s12175-007-0003-x, MR 2357811