# Algebraic structure

Important algebraic structures
Algebraic Axioms of the Group ring commutative
ring
Oblique body
(division ring)
body
Commutative law with regard to addition
Yes Yes Yes Yes
Distributive law Yes Yes Yes Yes
Commutative law with regard to multiplication
( multiplicative-commutative group )
No Yes No Yes
Multiplicative inverse exists
for every element except 0.
No No Yes Yes

The concept of algebraic structure (or universal algebra, general algebra or just algebra ) is a basic concept and central subject of investigation in the mathematical sub-area of universal algebra . An algebraic structure is usually a set with links on that set. A large number of the structures examined in abstract algebra such as groups , rings or solids are special algebraic structures.

Algebraic structures can also consist of several sets together with links to and between these sets. They are then called heterogeneous algebras , the most prominent example being vector spaces (with vectors and scalars).

Generalizations of algebraic structures are the partial algebras and the relational structures .

## Definition of the algebraic structure

An algebraic structure, or general algebra, is an ordered pair

${\ displaystyle {\ bigl (} A, (f_ {i}) _ {i \ in I} {\ bigr)},}$

consisting of a non-empty set of base amount or support amount of algebra, and a family of inner (last digit) links , and basic operations or fundamental operations referred to${\ displaystyle A,}$ ${\ displaystyle (f_ {i}) _ {i \ in I}}$${\ displaystyle A.}$

An inner - digit link${\ displaystyle n}$ to is a function that always maps elements from to an element from . A zero-digit link to can be interpreted as a clearly defined, distinguished element in a constant . Constants are usually identified with a special symbol (e.g. a letter or a number such as ). An inner single-digit link is a function from to which is often denoted by a symbol that is written immediately (i.e. without additional brackets or separators) before, after, over etc. the element (argument). ${\ displaystyle A}$ ${\ displaystyle f \ colon \, A ^ {n} \ to A,}$${\ displaystyle n}$${\ displaystyle a_ {1}, \ dotsc, a_ {n}}$${\ displaystyle A}$${\ displaystyle f (a_ {1}, \ dotsc, a_ {n})}$${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle A,}$${\ displaystyle e, 0.1}$${\ displaystyle A}$${\ displaystyle A,}$

Examples: ${\ displaystyle -a, \, a!, \, {\ overline {a}}, \, a ^ {- 1}}$

In the case of a two-digit link , the link symbol is usually written between the two arguments for the sake of simplicity.

Examples: instead of${\ displaystyle a + b, a \ cdot b, f \ circ g}$${\ displaystyle + (a, b), \ cdot (a, b), \ circ (f, g)}$

Most of the time an algebra only has a finite number of fundamental operations, one then just writes for the algebra${\ displaystyle f_ {1}, \ dotsc, f_ {m},}$${\ displaystyle (A, f_ {1}, \ dotsc, f_ {m}).}$

The (similarity) type (also signature ) of an algebra assigns the respective arity of the fundamental operation to each index , i.e. i.e., it is a function for The type can also be written as a family :${\ displaystyle {\ bigl (} A, (f_ {i}) _ {i \ in I} {\ bigr)}}$${\ displaystyle i \ in I}$ ${\ displaystyle n_ {i}}$${\ displaystyle f_ {i}}$${\ displaystyle \ sigma \ colon I \ to \ mathbb {N} _ {0}, \, i \ mapsto \ sigma (i): = n_ {i}}$${\ displaystyle f_ {i} \ colon \, A ^ {n_ {i}} \ to A.}$${\ displaystyle \ sigma = (n_ {i}) _ {i \ in I}.}$

For example, a group is usually understood as a structure , where the amount of carrier is, a two-digit link from to a constant in and a one-digit link from to. A group is thus an algebra of the type${\ displaystyle (G, \ cdot, 1, {} ^ {- 1})}$${\ displaystyle G}$${\ displaystyle \ cdot}$${\ displaystyle G \ times G}$${\ displaystyle G, 1}$${\ displaystyle G}$${\ displaystyle {} ^ {- 1}}$${\ displaystyle G}$${\ displaystyle G.}$${\ displaystyle (2,0,1).}$

### Remarks

• Sometimes it is also useful to allow the empty set as the carrier set of an algebra, for example to ensure that the set of all subalgebras (see below) of an algebra forms an algebraic lattice .${\ displaystyle A = \ emptyset}$
• Every non-empty set can be made into a trivial algebra with the identical mapping. Likewise, it can be regarded as an algebra with an empty family of links.${\ displaystyle A}$${\ displaystyle (A, \ operatorname {id})}$ ${\ displaystyle \ operatorname {id} \ colon A \ to A, a \ mapsto a.}$${\ displaystyle A}$${\ displaystyle (A, ())}$${\ displaystyle () = \ emptyset}$
• One could even allow "infinite-place algebras" with infinite-place combinations (e.g. σ-algebras ), but this would contradict the usual understanding of "algebraic".
• A generalization of general (complete) algebras are partial algebras in which not only total functions , but also partial functions are permitted as a link. For example, bodies are strictly speaking not complete algebras because only on is defined.${\ displaystyle (K, +, 0, -, \ cdot, 1, {} ^ {- 1})}$${\ displaystyle {} ^ {- 1}}$${\ displaystyle K \ setminus \ {0 \}}$

### Types of algebraic structures

The respective links of algebras of the same type often have common properties, so that algebras can be divided into different classes according to their type and the properties of their links . The properties of the concretely given links of an algebra are specified in more detail by axioms , which in abstract algebra (branch of mathematics) are usually written in the form of equations and which determine the type of algebra.

An example is the associative law for an inner two-digit link on a set${\ displaystyle *}$${\ displaystyle A \ colon}$

${\ displaystyle a * (b * c) = (a * b) * c}$for all elements of${\ displaystyle a, b, c}$${\ displaystyle A.}$

If the two-digit operation of an algebra fulfills this axiom ( replace through and through ), then the algebra belongs to the type of semigroup or it is a semigroup. ${\ displaystyle \ star}$${\ displaystyle (S, \ star)}$ ${\ displaystyle *}$${\ displaystyle \ star}$${\ displaystyle A}$${\ displaystyle S}$${\ displaystyle (S, \ star)}$

## Substructures (subalgebras)

If the basic set is an algebraic structure, one can define a new algebraic structure of the same type with the help of the links of on a subset of A, if the set is chosen so that the links of the original structure do not lead out of the set . This means that if you apply the links of the original algebraic structure to the elements of , no elements may arise that are not in - in particular, the constants must already be contained in. In the specific application z. B. Subgroups the substructures of a group. Depending on how you have chosen the equations for defining the algebraic structure, the substructures can look completely different. So z. B. Define groups in such a way that the substructures are normal dividers . ${\ displaystyle A}$${\ displaystyle A}$${\ displaystyle B \ subseteq A}$${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle B}$${\ displaystyle B}$

## Homomorphisms

Structurally accurate mappings, so-called homomorphisms , between two algebraic structures and of the same type (that is, they have links of the same arithmetic and given specific properties) are compatible with the links between the two algebraic structures . Every algebraic structure therefore has its own homomorphism concept and therefore defines a category . ${\ displaystyle A}$${\ displaystyle B}$

Corresponding links in and are usually denoted by the same symbol. For example, in each considered group, the group operation is uniformly z. B. written. If the two links have to be kept apart in individual cases, the symbols of their basic sets or the like are usually added as indices, e.g. B. and . A homomorphism is a function that fulfills the following condition for every link (with arity ): ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle \ cdot}$${\ displaystyle \ cdot _ {A}}$${\ displaystyle \ cdot _ {B}}$${\ displaystyle \ varphi \ colon A \ to B}$${\ displaystyle f}$${\ displaystyle n}$

${\ displaystyle \ varphi (f_ {A} (x_ {1}, \ dotsc, x_ {n})) = f_ {B} (\ varphi (x_ {1}), \ dotsc, \ varphi (x_ {n} ))}$

The special notations of the zero, one and two-digit links are taken into account:

• If each is the constants of zero-digit links, then is${\ displaystyle k_ {A}, k_ {B}}$${\ displaystyle \ varphi (k_ {A}) = k_ {B}.}$
• If there is a single-digit link , then a single-digit link can also be written as an exponent, index, etc. B. with and with then${\ displaystyle -}$${\ displaystyle \ varphi (- (x)) = - (\ varphi (x)).}$${\ displaystyle x ^ {- 1}: = {} ^ {- 1} (x) \, \!}$${\ displaystyle \ varphi (x) ^ {- 1}: = {} ^ {- 1} (\ varphi (x)) \, \!}$${\ displaystyle \ varphi (x ^ {- 1}) = \ varphi (x) ^ {- 1}.}$
• For two-digit links is${\ displaystyle +}$${\ displaystyle \ varphi (x_ {1} + x_ {2}) = \ varphi (x_ {1}) + \ varphi (x_ {2}).}$

A surjective homomorphism is called an epimorphism , an injective monomorphism . A homomorphism of in itself (i.e. if applies) is called an endomorphism . A bijective homomorphism whose inverse function is also a homomorphism is called an isomorphism . If the isomorphism is also endomorphism, it is called automorphism .${\ displaystyle A}$${\ displaystyle B = A}$

## Congruence relations

On algebraic structures, special types of equivalence relations can be found that are compatible with the links of an algebraic structure . These are then called congruence relations . With the help of congruence relations, factor algebras can be formed, i. This means that a structure of the same type is generated from the original algebraic structure, the elements of which, however, are then the equivalence classes with regard to the congruence relation. The links are well-defined due to the special properties of the congruence relation . In many specific applications, the equivalence classes correspond to the secondary or congruence classes of certain substructures, e.g. B. normal dividers for groups or ideals for rings etc.

## Products

If one forms the set-theoretical direct product of the basic sets of several general algebras of the same type, one can in turn obtain a new algebra of the same type on this product set by defining the new links of this algebra component by component through the links of the original algebras. However, this can have different properties than the original algebra; z. B. the product of bodies no longer has to be a body.

For a generalization of the direct product of algebras see: Subdirect product . There Birkhoff's theorem is presented, according to which every algebra is a subdirect product of subdirectly irreducible algebras.

## "Zoo" of algebraic structures

### Example: groups

As an example of defining an algebraic structure, consider a group. Usually, a group is defined as a pair consisting of a quantity and a two-digit shortcut that for all in are satisfied the following three axioms: ${\ displaystyle (G, *),}$${\ displaystyle G}$${\ displaystyle *,}$${\ displaystyle x, y, z}$${\ displaystyle G}$

• ${\ displaystyle x * (y * z) = (x * y) * z}$( Associativity ).
• There is an in such that ( neutral element ).${\ displaystyle e}$${\ displaystyle G}$${\ displaystyle e * x = x = x * e}$
• For each there is an in such that ( inverse element ).${\ displaystyle x}$${\ displaystyle i}$${\ displaystyle G}$${\ displaystyle x * i = e = i * x}$

Sometimes one still finds the requirement of “seclusion” that should lie in again , but from the point of view of an algebraic the term “two-digit link” already includes this property. ${\ displaystyle x * y}$${\ displaystyle G}$

However, this definition has the property that the axioms are not only expressed by equations, but also contain the existential quantifier "there is ... so that"; in general algebra one tries to avoid such axioms ( quantifier elimination ). The simplification of the axioms to a pure equation form is not difficult here: we add a zero-digit link and a one-digit link and define a group as a quadruple with a set of a two-digit link of a constant and a one-digit link that satisfy the following axioms: ${\ displaystyle e}$${\ displaystyle {} ^ {- 1}}$ ${\ displaystyle (G, *, e, ^ {- 1})}$${\ displaystyle G,}$${\ displaystyle *,}$${\ displaystyle e}$${\ displaystyle {} ^ {- 1}}$

• ${\ displaystyle x * (y * z) = (x * y) * z}$
• ${\ displaystyle e * x = x = x * e}$
• ${\ displaystyle x * x ^ {- 1} = e = x ^ {- 1} * x}$

It is now important to check whether the definition of a group has actually been achieved. It could be that not all properties of a group are given or even too many. In fact, the two definitions of a group are the same.

### Examples of algebraic structures

The following list shows all (two-digit) links, neutral elements (= zero-digit links), inverse mappings (= single-digit links) and operator ranges.

In normal use, on the other hand, for algebraic structures, only the multi-character links and the operator ranges are given (sometimes the neutral elements), for all others there are usually standard notations.

A non-exhaustive list of various algebraic structures:

• Gruppoid or Magma , also binary or operational : a non-empty set with a two-digit link .${\ displaystyle (O, *)}$${\ displaystyle O}$ ${\ displaystyle *}$
• Semigroup : an associative groupoid.${\ displaystyle (S, *)}$
• Semi-lattice : a commutative semi-group in which every element is idempotent .${\ displaystyle (S, *)}$
• Monoid : a semigroup with a neutral element .${\ displaystyle (M, *, e)}$ ${\ displaystyle e}$
• Group : a monoid with an inverse element for each element .${\ displaystyle (G, *, e, ^ {- 1})}$ ${\ displaystyle a ^ {- 1}}$${\ displaystyle a}$
• Abelian group : a commutative group. Abelian groups are preferably written additively and then called modules, the inverse of an element is now called the opposite .${\ displaystyle (G, *, e, ^ {- 1})}$${\ displaystyle (G, +, 0, -)}$${\ displaystyle a}$ ${\ displaystyle -a}$
• Half ring : a set with two links ( addition ) and ( multiplication ), with which and are semigroups and the distributive laws are fulfilled. Often, however , it should also be commutative and / or have a neutral element 0, the zero element of the half-ring: The definitions are not uniform here!${\ displaystyle (H, +, \ cdot)}$${\ displaystyle H}$${\ displaystyle +}$${\ displaystyle \ cdot}$${\ displaystyle (H, +)}$${\ displaystyle (H, \ cdot)}$${\ displaystyle (H, +)}$
• Lattice : a set with two links (union) and (intersection), so that and are commutative semigroups and the laws of absorption are fulfilled. and are then semi-associations.${\ displaystyle (V, \ vee, \ wedge)}$${\ displaystyle V}$${\ displaystyle \ vee}$${\ displaystyle \ wedge}$${\ displaystyle (V, \ vee)}$${\ displaystyle (V, \ wedge)}$${\ displaystyle (V, \ vee)}$${\ displaystyle (V, \ wedge)}$
• Boolean lattice or Boolean algebra : and are commutative monoids, is a half ring and every element has a complement .${\ displaystyle (B, \ vee, 0, \ wedge, 1, \ neg)}$${\ displaystyle (B, \ vee, 0)}$${\ displaystyle (B, \ wedge, 1)}$${\ displaystyle (B, \ vee, \ wedge)}$${\ displaystyle a}$ ${\ displaystyle \ neg a}$
• Ring : is an Abelian group and a half ring.${\ displaystyle (R, +, 0, -, \ cdot)}$${\ displaystyle (R, +, 0, -)}$${\ displaystyle (R, +, \ cdot)}$

## Provided with further structure, internalization

Algebraic structures can be equipped with additional structures, e.g. B. with a topology . A topological group is a topological space with a group structure so that the operations multiplication and inverse formation are continuous . A topological group has both a topological and an algebraic structure. Other common examples are topological vector spaces and Lie groups . In abstract terms, the links in such structures are morphisms in a certain category , such as that of the topological spaces in the case of topological groups. One speaks of internalization in this category. In the special case of common algebraic structures, the links are morphisms in the category of sets, i.e. functions.

## Partial algebras

If the term links in the above definition is replaced by partial links, then one speaks of a partial algebra. The links do not have to be defined here for all combinations of parameters ( tuple combinations). ${\ displaystyle n}$

## literature

1. The index set can be understood as an alphabet of identifiers for the functions. The couple is then sometimes referred to as the signature .${\ displaystyle I}$${\ displaystyle (I, \ sigma)}$