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 [math]\displaystyle{ X }[/math] with a constant [math]\displaystyle{ 0 }[/math] and a binary operation [math]\displaystyle{ * }[/math] satisfying the following:
- [math]\displaystyle{ x*x=0 }[/math]
- [math]\displaystyle{ x*0=x }[/math]
- [math]\displaystyle{ 0*(x*y)=y*x }[/math]
Example
Let [math]\displaystyle{ Z }[/math] be the set of integers and '[math]\displaystyle{ - }[/math]' be the binary operation 'subtraction'. Then the algebraic structure [math]\displaystyle{ (Z,-) }[/math] obeys the following properties:
- [math]\displaystyle{ x-x=0 }[/math]
- [math]\displaystyle{ x-0=x }[/math]
- [math]\displaystyle{ 0-(x-y)=y-x }[/math]
References
- Walendziak, Andrzej (2007), "On BF-algebras", Math. Slovaca 57 (2): 119–128, doi:10.2478/s12175-007-0003-x
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