Additive map

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In algebra, an additive map or additive function is a function f {\displaystyle f} {\displaystyle f} that preserves the addition operation:[1] f ( x + y ) = f ( x ) + f ( y ) {\displaystyle f(x+y)=f(x)+f(y)} {\displaystyle f(x+y)=f(x)+f(y)} for every pair of elements x {\displaystyle x} {\displaystyle x} and y {\displaystyle y} {\displaystyle y} in the domain of ⁠ f {\displaystyle f} {\displaystyle f}⁠. For example, any linear map is additive. When the domain is the integers (ℤ), this is called a Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }-linear map. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial.

More formally, an additive map is a Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }-module homomorphism. Since an abelian group is a Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }-module, it may be defined as a group homomorphism between abelian groups.

A map V × W → X {\displaystyle V\times W\to X} {\displaystyle V\times W\to X} that is additive in each of two arguments separately is called a bi-additive map or a Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }-bilinear map.[2]

Examples

Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring.

If f {\displaystyle f} {\displaystyle f} and g {\displaystyle g} {\displaystyle g} are additive maps, then the map f + g {\displaystyle f+g} {\displaystyle f+g} (defined pointwise) is additive.

Properties

Definition of scalar multiplication by an integer

Suppose that X {\displaystyle X} {\displaystyle X} is an additive group with identity element 0 {\displaystyle 0} {\displaystyle 0} and that the inverse of x ∈ X {\displaystyle x\in X} {\displaystyle x\in X} is denoted by ⁠ − x {\displaystyle -x} {\displaystyle -x}⁠. For any x ∈ X {\displaystyle x\in X} {\displaystyle x\in X} and integer ⁠ n ∈ Z {\displaystyle n\in \mathbb {Z} } {\displaystyle n\in \mathbb {Z} }⁠, let: n x := { 0            when  n = 0 x + ⋯ + x         ( n  summands)     when  n > 0 ( − x ) + ⋯ + ( − x )         ( | n |  summands)     when  n < 0. {\displaystyle nx:=\left\{{\begin{alignedat}{9}&&&0&&&&&&~~~~&&&&~{\text{ when }}n=0\\&&&x&&+\cdots +&&x&&~~~~{\text{(}}n&&{\text{ summands) }}&&~{\text{ when }}n>0\\&(-&&x)&&+\cdots +(-&&x)&&~~~~{\text{(}}|n|&&{\text{ summands) }}&&~{\text{ when }}n<0.\\\end{alignedat}}\right.} {\displaystyle nx:=\left\{{\begin{alignedat}{9}&&&0&&&&&&~~~~&&&&~{\text{ when }}n=0\\&&&x&&+\cdots +&&x&&~~~~{\text{(}}n&&{\text{ summands) }}&&~{\text{ when }}n>0\\&(-&&x)&&+\cdots +(-&&x)&&~~~~{\text{(}}|n|&&{\text{ summands) }}&&~{\text{ when }}n<0.\\\end{alignedat}}\right.} Thus ( − 1 ) x = − x {\displaystyle (-1)x=-x} {\displaystyle (-1)x=-x} and it can be shown that for all integers m , n ∈ Z {\displaystyle m,n\in \mathbb {Z} } {\displaystyle m,n\in \mathbb {Z} } and all ⁠ x ∈ X {\displaystyle x\in X} {\displaystyle x\in X}⁠, ( m + n ) x = m x + n x {\displaystyle (m+n)x=mx+nx} {\displaystyle (m+n)x=mx+nx} and ⁠ − ( n x ) = ( − n ) x = n ( − x ) {\displaystyle -(nx)=(-n)x=n(-x)} {\displaystyle -(nx)=(-n)x=n(-x)}⁠. This definition of scalar multiplication makes the cyclic subgroup Z x {\displaystyle \mathbb {Z} x} {\displaystyle \mathbb {Z} x} of X {\displaystyle X} {\displaystyle X} into a left Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }-module; if X {\displaystyle X} {\displaystyle X} is commutative, then it also makes X {\displaystyle X} {\displaystyle X} into a left Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }-module.

Homogeneity over the integers

If f : X → Y {\displaystyle f:X\to Y} {\displaystyle f:X\to Y} is an additive map between additive groups then f ( 0 ) = 0 {\displaystyle f(0)=0} {\displaystyle f(0)=0} and for all ⁠ x ∈ X {\displaystyle x\in X} {\displaystyle x\in X}⁠, f ( − x ) = − f ( x ) {\displaystyle f(-x)=-f(x)} {\displaystyle f(-x)=-f(x)} (where negation denotes the additive inverse) and[proof 1] f ( n x ) = n f ( x )  for all  n ∈ Z . {\displaystyle f(nx)=nf(x)\quad {\text{ for all }}n\in \mathbb {Z} .} {\displaystyle f(nx)=nf(x)\quad {\text{ for all }}n\in \mathbb {Z} .} Consequently, f ( x − y ) = f ( x ) − f ( y ) {\displaystyle f(x-y)=f(x)-f(y)} {\displaystyle f(x-y)=f(x)-f(y)} for all x , y ∈ X {\displaystyle x,y\in X} {\displaystyle x,y\in X} (where, by definition, ⁠ x − y := x + ( − y ) {\displaystyle x-y:=x+(-y)} {\displaystyle x-y:=x+(-y)}⁠).

In other words, every additive map is homogeneous over the integers. Consequently, every additive map between abelian groups is a homomorphism of Z {\displaystyle \mathbb {Z} } {\displaystyle \mathbb {Z} }-modules.

Homomorphism of Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} }-modules

If the additive abelian groups X {\displaystyle X} {\displaystyle X} and Y {\displaystyle Y} {\displaystyle Y} are also a unital modules over the rationals Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} } (such as real or complex vector spaces) then an additive map f : X → Y {\displaystyle f:X\to Y} {\displaystyle f:X\to Y} satisfies:[proof 2] f ( q x ) = q f ( x )  for all  q ∈ Q  and  x ∈ X . {\displaystyle f(qx)=qf(x)\quad {\text{ for all }}q\in \mathbb {Q} {\text{ and }}x\in X.} {\displaystyle f(qx)=qf(x)\quad {\text{ for all }}q\in \mathbb {Q} {\text{ and }}x\in X.} In other words, every additive map is homogeneous over the rational numbers. Consequently, every additive maps between unital Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} }-modules is a homomorphism of Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} }-modules.

Despite being homogeneous over ⁠ Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} }⁠, as described in the article on Cauchy's functional equation, even when ⁠ X = Y = R {\displaystyle X=Y=\mathbb {R} } {\displaystyle X=Y=\mathbb {R} }⁠, it is nevertheless still possible for the additive function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } {\displaystyle f:\mathbb {R} \to \mathbb {R} } to not be homogeneous over the real numbers; said differently, there exist additive maps f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } {\displaystyle f:\mathbb {R} \to \mathbb {R} } that are not of the form f ( x ) = s 0 x {\displaystyle f(x)=s_{0}x} {\displaystyle f(x)=s_{0}x} for some constant ⁠ s 0 ∈ R {\displaystyle s_{0}\in \mathbb {R} } {\displaystyle s_{0}\in \mathbb {R} }⁠. In particular, there exist additive maps that are not linear maps with respect to an existing ring structure of the codomain.

See also

Notes

  1. Leslie Hogben (2013), Handbook of Linear Algebra (3 ed.), CRC Press, p. 30–8, ISBN 9781498785600
  2. N. Bourbaki (1989), Algebra Chapters 1–3, Springer, p. 243

Proofs

  1. f ( 0 ) = f ( 0 + 0 ) = f ( 0 ) + f ( 0 ) {\displaystyle f(0)=f(0+0)=f(0)+f(0)} {\displaystyle f(0)=f(0+0)=f(0)+f(0)} so adding − f ( 0 ) {\displaystyle -f(0)} {\displaystyle -f(0)} to both sides proves that ⁠ f ( 0 ) = 0 {\displaystyle f(0)=0} {\displaystyle f(0)=0}⁠. If x ∈ X {\displaystyle x\in X} {\displaystyle x\in X} then 0 = f ( 0 ) = f ( x + ( − x ) ) = f ( x ) + f ( − x ) {\displaystyle 0=f(0)=f(x+(-x))=f(x)+f(-x)} {\displaystyle 0=f(0)=f(x+(-x))=f(x)+f(-x)} so that f ( − x ) = − f ( x ) {\displaystyle f(-x)=-f(x)} {\displaystyle f(-x)=-f(x)} where, by definition, ⁠ ( − 1 ) f ( x ) := − f ( x ) {\displaystyle (-1)f(x):=-f(x)} {\displaystyle (-1)f(x):=-f(x)}⁠. Induction shows that if n ∈ N {\displaystyle n\in \mathbb {N} } {\displaystyle n\in \mathbb {N} } is positive then f ( n x ) = n f ( x ) {\displaystyle f(nx)=nf(x)} {\displaystyle f(nx)=nf(x)} and that the additive inverse of n f ( x ) {\displaystyle nf(x)} {\displaystyle nf(x)} is ⁠ n ( − f ( x ) ) {\displaystyle n(-f(x))} {\displaystyle n(-f(x))}⁠, which implies that f ( ( − n ) x ) = f ( n ( − x ) ) = n f ( − x ) = n ( − f ( x ) ) = − ( n f ( x ) ) = ( − n ) f ( x ) {\displaystyle f((-n)x)=f(n(-x))=nf(-x)=n(-f(x))=-(nf(x))=(-n)f(x)} {\displaystyle f((-n)x)=f(n(-x))=nf(-x)=n(-f(x))=-(nf(x))=(-n)f(x)} (this shows that f ( n x ) = n f ( x ) {\displaystyle f(nx)=nf(x)} {\displaystyle f(nx)=nf(x)} holds for ⁠ n < 0 {\displaystyle n<0} {\displaystyle n<0}⁠). ◼ {\displaystyle \blacksquare } {\displaystyle \blacksquare }
  2. Let x ∈ X {\displaystyle x\in X} {\displaystyle x\in X} and q = m n ∈ Q {\displaystyle q={\frac {m}{n}}\in \mathbb {Q} } {\displaystyle q={\frac {m}{n}}\in \mathbb {Q} } where m , n ∈ Z {\displaystyle m,n\in \mathbb {Z} } {\displaystyle m,n\in \mathbb {Z} } and ⁠ n > 0 {\displaystyle n>0} {\displaystyle n>0}⁠. Let ⁠ y := 1 n x {\displaystyle y:={\frac {1}{n}}x} {\displaystyle y:={\frac {1}{n}}x}⁠. Then ⁠ n y = n ( 1 n x ) = ( n 1 n ) x = ( 1 ) x = x {\displaystyle ny=n\left({\frac {1}{n}}x\right)=\left(n{\frac {1}{n}}\right)x=(1)x=x} {\displaystyle ny=n\left({\frac {1}{n}}x\right)=\left(n{\frac {1}{n}}\right)x=(1)x=x}⁠, which implies f ( x ) = f ( n y ) = n f ( y ) = n f ( 1 n x ) {\displaystyle f(x)=f(ny)=nf(y)=nf\left({\frac {1}{n}}x\right)} {\displaystyle f(x)=f(ny)=nf(y)=nf\left({\frac {1}{n}}x\right)} so that multiplying both sides by 1 n {\displaystyle {\frac {1}{n}}} {\displaystyle {\frac {1}{n}}} proves that ⁠ f ( 1 n x ) = 1 n f ( x ) {\displaystyle f\left({\frac {1}{n}}x\right)={\frac {1}{n}}f(x)} {\displaystyle f\left({\frac {1}{n}}x\right)={\frac {1}{n}}f(x)}⁠. Consequently, ⁠ f ( q x ) = f ( m n x ) = m f ( 1 n x ) = m ( 1 n f ( x ) ) = q f ( x ) {\displaystyle f(qx)=f\left({\frac {m}{n}}x\right)=mf\left({\frac {1}{n}}x\right)=m\left({\frac {1}{n}}f(x)\right)=qf(x)} {\displaystyle f(qx)=f\left({\frac {m}{n}}x\right)=mf\left({\frac {1}{n}}x\right)=m\left({\frac {1}{n}}f(x)\right)=qf(x)}⁠. ◼ {\displaystyle \blacksquare } {\displaystyle \blacksquare }

References