4.3 Mappings

While the algebraic structures focus on the interplay of elements within given sets, a means to assign elements from differing sets as a special yet highly important case of Definition 4 is desired.

Definition 20 (Mapping) Considering a relation f , between the sets 𝒟 and ℬ , it is considered a map, mapping, or function, if every element of 𝒟 is assigned to an element of ℬ . The converse, however, is not necessarily true that is an element of ℬ , called an image, may result from several elements of 𝒟 , called preimages. The set 𝒟 is called the domain, the set ℬ the codomain of the function f .

The terms mapping and function can be used synonymously.

Similar to the qualification of relations, mappings can also be further qualified. Qualifications regarding the domain and codomain are provided, from which can be deduced, if a mapping is invertible.

Definition 21 (Injective) A mapping

f : 𝒜 → ℬ                                  (4.20)
is called injective, if there is at most one value b = f(a) ∈ ℬ for each element a ∈ 𝒜 . This means that every preimage a ∈ 𝒜 has an image b ∈ ℬ , but there may be elements in ℬ , which are not obtained as images.

Connected to injectivity is the concept of a kernel:

Definition 22 (Kernel) For a given mapping

f : 𝒜 → ℬ                                  (4.21)
the set of elements resulting in the identity element e ∈ ℬ is called the kernel of the mapping f
kerf =  {x ∈ 𝒜 |f(x) = e ∈ ℬ}                         (4 .22)

As such the kernel provides information on the injectivity of a mapping. If a mapping is injective, the kernel is trivial

kerf =  ∅.                                 (4.23)

Where injectivity states that there is at most one solution to every element from the domain of a function, it is also possible to assert that there is at least one image for each preimage, as is done in the next definition.

Definition 23 (Surjective) A mapping

f : 𝒜 → ℬ                                  (4.24)
is called surjective or onto, if b = f(a)  has at least one solution b ∈ ℬ for every a ∈ 𝒜 . Several identical images in 𝒜 may have different preimages in ℬ .

The combination of Definition 21 and Definition 23 yields:

Definition 24 (Bijective) A mapping

f : 𝒜 → ℬ                                  (4.25)
that is both injective and surjective is called bijective. It maps every preimage from ℬ to a distinct image in 𝒜 . There are no elements in 𝒜 which cannot be obtained as images.

A bijective mapping, also called a bijection, is an invertible function. Since injectivity (Definition 21) demands the triviality of the kernel (Definition 22), it follows that an invertible function necessarily also has a trivial kernel.

Mappings are an essential instrument in mathematical descriptions and modelling, because it is possible to compose individual mappings in order to construct new mappings. Considering the mappings

𝒜 ----- ℬ                                  (4.26)

it is possible to combine them in a fashion as to obtain:
g ∘ f : 𝒜 → 𝒞                                (4.27)
   𝒜  -----ℬ|                                (4.28)
This may also be presented as
∀x ∈ 𝒜  →  g(f(x)) ∈ 𝒞                           (4.29)
more constructively illustrating the procedure, how the new mapping is to be constructed. The agreement of the codomain of f and the domain of g are crucial for the feasibility of such a composition.

Mappings and their compositions are essential for the operation of modern computational machines, e.g., when dealing with memory management such as virtual memory, which relies on partitions as well as mappings.

The definition of cases of mappings simplifies the formulation of subsequent definitions. Therefore, several special cases are provided here in a collected form.

Definition 25 (Projection map) A projection map projj  , which may also given as πj  , extracts the j th component of elements from a Cartesian product (Definition 3) space x ∈ X  × ... × X  × ...×  X
      1          j          k  to Xj :

proj (x) = πj(x ) = xj                           (4.30)

A mapping, which can be used to extend a concept from a setting to another, as is seen in Section 5.2, is provided in the following.

Definition 26 (Lift) Given the mappings f : 𝒜 →  ℬ and g : 𝒞 → ℬ , a lift is defined as a map h with the property that g ∘ h = f

  f -----
𝒜  h      ℬ|                                (4.31)

A subsequently important class of mappings can be abstractly described here, where realizations are then provided by Definition 70 and Definition 68.

Definition 27 (Derivation) Given an associative algebra 𝒜 (Definition 17) and a module ℳ (Definition 15), a derivation is defined as a mapping

        D : 𝒜 →  ℳ  w ith the property                   (4.32a)

D(f g) = D (f)g + fD (g)    f, g ∈ 𝒜                   (4.32b)