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BFunktion

BFunktion is a term used in mathematics and theoretical computer science to denote a parametric family of binary functions. For a nonempty set X and a parameter set B, the BFunktion consists of functions f_b: X × X → X for each b ∈ B. The parameter b selects a specific operation within the family, allowing a single notation to cover many related binary operations. In algebraic terms, BFunktion can be described by a single function φ: B × X × X → X with f_b(x,y) = φ(b,x,y).

Common examples include the projection family, where f_0(x,y) = x and f_1(x,y) = y, and the weighted combination

BFunktion is used in contexts where a single framework is needed to describe multiple binary operations. Applications

See also: binary operation; parameterized function; family of operations; interpolation operator.

f_b(x,y)
=
(1
−
b)
x
+
b
y
for
x,y
in
a
vector
space
X
and
b
in
[0,1].
Depending
on
X
and
φ,
certain
parameter
values
yield
associative,
commutative,
or
idempotent
operations,
while
others
may
lack
these
properties.
appear
in
data
fusion,
interpolation
and
blending
in
graphics,
and
the
modeling
of
choice
operators
in
programming
languages.
Researchers
study
how
the
parameterization
affects
algebraic
properties,
identities,
and
the
behavior
of
composition
within
the
family.