4.2 Algebraic Structures

Beside the bare structures described so far, the structures resulting from operations defined on top of the sets as a combined union gives rise to algebraic structures. They are of importance as they provide rules, how elements interacts and create new elements.

In the setting of digital computers types are models or extensions of algebraic structures. Therefore, several algebraic structures are introduced in the following with an increasing number of requirements imposed on elements and/or operations.

Definition 10 (Monoid) A set of elements M on which a binary operation (⋅)  is defined is called a monoid (M,⋅)  , if the binary operation satisfies the following conditions:

Further requirements on Definition 10 result in

Definition 11 (Group) A monoid (G,⋅)  , where the binary operation additionally fulfils the condition that for every element in G an inverse element exists, which produces the identity element under the binary operation is called a group

a ⋅ b = e,   ∀a, b ∈ G                           (4.12)

and as a further qualification

Definition 12 (Abelian group) When the order of the operands of the binary operation (⋅)  of a group (G,⋅) does not change its result, the group is called Abelian or commutative.

a ⋅ b = b ⋅ a, ∀a, b ∈ G                         (4.13)

Groups are basic building blocks in the exploration of further structures, which can be defined by demanding additional operations on the basic sets.

Definition 13 (Ring) An Abelian group (Definition 12), which is equipped with an additional binary operation under which it is a monoid (Definition 10), is called a ring (R,+, ⋅)  , if the two binary relations are distributive:

a(b + c) = (ab) + (ac)                                  (4.14)
(a + b)c = (ac) + (bc)    ∀a, b,c ∈ R                    (4.15)

The structure of the described ring is, however, insufficient to describe the basic notion of real numbers ℝ or the complex numbers ℂ  . To this end the following definition is required:

Definition 14 (Field) If the multiplication operation of a ring K is invertible ∀x ∈ K  ∖ {0} , it is called a field.

The next definition describes the algebraic structure of entities (Definition 61), which have proven to be immensely useful.

Definition 15 (Module) A Module ℳ over a ring R  (Definition 13) is an Abelian group (Definition 12) with respect to the operation of the addition of two elements u,v ∈ 𝒱 , while additionally being a ring with respect to the operation of multiplying elements v ∈ 𝒱 by elements a∈R , which are called scalars.

A related definition with somewhat stronger requirements yields a structure, which is essential for the construction of simple geometric settings.

Definition 16 (Vector space) A vector space 𝒱 over a field 𝔽  is an Abelian group with respect to the operation of the addition of two elements u, v ∈ 𝒱 , while additionally being a ring with respect to the operation of multiplying elements v ∈ 𝒱 by elements a ∈ 𝔽  . Elements of a vector space v ∈ 𝒱 are called vectors.

While the elements of vector spaces, the vectors, constitute a powerful concept, they are insufficient to describe all the entities required in modelling scientific processes. Additional entities are therefore required. It is not limited to entities introduced later and therefore provided here to clearly distinguish the algebraic structure from the elements.

Definition 17 (Algebra) An algebra A over a field F is a vector space equipped with an additional binary relation (Definition 4)

⋅ : A × A → A                                (4.16)

A further qualification of the just defined structure may be possible. The availability of the following term allows a more precise classification as found in Definition 56 and in conjunction with Definition 60.

Definition 18 (Graded algebra) In case the algebra admits the decomposition into additive groups of the form

A =      An,                                 (4 .17)
where additionally the multiplication operation results in
Ai × Aj →  Ai+j,                               (4 .18)
it is called a graded algebra.

Among the most versatile and useful, almost ubiquitous, algebraic entities are:

Definition 19 (Polynomials) A formal prescription of the form

p(X ) =     ciXi,      ci ∈ R, i ∈ ℕ ∪ {0 }                (4.19)
with coefficients ci  is called a polynomial over the ring (Definition 13) R in the variable X .

The variable X in the purely algebraic definition is a formal symbol and need not be an element of a field (Definition 14), such as ℝ  or ℂ  , as in the case of polynomial codes. The algebraic considerations, however, assert that polynomials defined in this fashion can be added (subtracted) and multiplied, thus forming a ring (Definition 13). The case that the variables X are either from ℝ  or ℂ is of particular usefulness in many fields of mathematics, with the field of interpolation as well integration among them. Then the expression   i
X  is simply the i th power of a variable x∈ℝ(ℂ )  and values can be derived by simple multiplication within their respective fields.