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 $f$ .

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.

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

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 $ℬ$ .

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

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) j

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) | | 𝒞g

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)

4.3 Mappings