BF-algebra

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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

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

1. ${\displaystyle x*x=0}$
2. ${\displaystyle x*0=x}$
3. ${\displaystyle 0*(x*y)=y*x}$

Example

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

1. ${\displaystyle x-x=0}$
2. ${\displaystyle x-0=x}$
3. ${\displaystyle 0-(x-y)=y-x}$

References

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