Generator (mathematics)
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The generating set is a term on the mathematics that is used in particular in the field of algebra. The generating set is a subset of a mathematical space, so a lot with a structure. Through a structure acceptable design may consist of the generating set is a subset of the output space to create. This generated subset is called the product of the predetermined amount or the system of generators in the considered space.
With the structure compatibility means that the axioms that apply to the room with its structure, also apply to a subset with the correspondingly restricted structure. Such a subset is called then a subspace.
Examples of products are
- The set of linear combinations of a set of generators in a vector space (see Linear Case ).
- The topological product of a subset of a topological space in this topological space.
- The product of a subset of a group in this group.
However, at a given product, the generating set is not uniquely determined. The existence of a system of generators is usually easy to show, as you can often select the product itself as a system of generators. however, this is rarely useful. Often an attempt is made to select the minimal set of generators. However, this is not always possible and general existence proofs for minimal systems of generators do not rarely from Zorn's lemma use ( see, eg, existence of a basis in vector spaces ).
- 3.1 Vector Spaces
- 3.2 groups
- 3.3 Topological Groups
- 3.4 Rings
- 3.5 equivalence relations
- 3.6 congruence
- 3.7 topologies
- 3.8 σ - algebras
Construction method
By means of a system of generators can be a mathematical space konstruiiert. There are two generally different methods.
One is to the average of all subspaces, which comprise the system of generators look at. Since the generating set is a subset of the considered space, the space itself is always a subspace, which comprises the set of generators. This is important so that the definition makes sense, since the average over an empty set is not a set itself.
In the other construction method is considered the number of possible combinations of structural elements of the generating system. In the vector space as the set of linear combinations of elements of the generating set.
Concrete examples
The group of integers
An illustrative example is the group with the neutral element 0 The allowed operations are addition and here the transition to a negative number, because these are precisely the group operations.
The empty set generates the trivial subgroup, because of a subset demanded you that it contains the neutral element and then the smallest subgroup that contains the empty set, because it is the smallest subgroup at all.
Is generated as a group by, that is is a producer of, for every positive number can win and all others as 1 (-1 ) ... ( by successive addition of 1 ... 1 from the 1 - 1).
In this case, the system of generators is minimal, because the only proper subset is the empty set, and this is not a system of generators for. Another set of generators is because the product contains 1 = 3 ( -2) and therefore quite, since we already know that one is a producer. It is even minimal, ie no proper subset of E is a generating system. This example shows that minimal systems of generators need not necessarily be of minimal cardinality, because is a generating system of genuine smaller cardinality.
Generally produced by a non-empty subset, if the greatest common divisor of all the elements from the amount has. This shows the Euclidean algorithm, because this produced as a by- product of a representation as a whole linear combination of elements ( and any such linear combination of split ).
General examples
Vector spaces
A set of vectors of a vector space is called system of generators of the vector space, if each element is represented as a linear combination of vectors from the set:
Is now a vector space is given, you can look for the smallest number of vectors ask what produce. A minimal generating system exists in this case and is called a basis of the vector space, the cardinality of a basis is the dimension of the vector space on.
Since the intersection of a non-empty set of sub- areas is in turn sub-space of, and a lower space ( self ) which contains, can be considered the average of all sub-spaces of which contain. This is clearly the smallest sub-space within the meaning of the inclusion, which has the property of containing a subset. It is not difficult to show that this subspace of the generated within the meaning of the previous definition is accurate ( ie there as all possible linear combinations of element ).
Groups
In the case of a group the subgroup generated by a subset is often referred to. Applies, it is said to be generated by the crowd. Does the group have called a finite generating system so the group finally produced.
Clearly contains the neutral element of, and finite products for which it holds for each or.
In particular, is a singleton, ie, we write instead of and also called cyclic with generator. This applies, ie, the product consists of the integer powers of the generator.
Generally, the product is the image under the canonical mapping of the free group on the set, the inclusion continues. This explains the above explicit description of the product. Furthermore, this interpretation finds important applications in group theory. We assume that is surjective, ie that is generated by. The knowledge of the core by then determines uniquely up to isomorphism. In favorable cases, the core can in turn easily described by producers. Date is then up to isomorphism uniquely determined.
Topological groups
In the theory of topological groups we are interested in the rule for closed subgroups and therefore agreed, under the product of a subset to understand the smallest closed subgroup containing.
Since the links and the inverses are continuous, the conclusion of the algebraic product is again a subgroup of. Therefore the product is a subset of a topological group of the completion of the group product.
Own as a topological group is a finite set of generators, it is referred to as topologically finitely generated.
As in all p- adic numbers is dense, is generated as a topological group. So it is topologically finitely generated. From the terminology of profinite groups infers that is pro-cyclical.
Rings
As a ring with unity generated by the empty set. This reflects the fact that the initial object in the category of rings with unity.
Be a commutative ring with 1 A system of generators is an ideal of a set with the property that each can decompose as with. An ideal is called finitely generated if there is a finite subset with. A principal ideal is generated by a one - element set ideal. In particular, the ring R is a principal ideal, because it is generated from. A ring is Noetherian if and only if all ideals are finitely generated.
Equivalence relations
Equivalence relations are sometimes difficult to describe explicitly. Often one would like to construct an equivalence relation that identifies certain predefined elements and should also receives certain properties, for example, is compatible with predefined shortcuts (that is a congruence ).
Be a lot and given an arbitrary relation. Then the equivalence relation generated by can also be described by saying that if and only if
- Or
- There are a finite number of elements, and for each or.
The explicit indication in this case it is relatively complicated.
Congruence
The above concept is used in particular for the construction of normal subgroups and ideals or general congruence relations.
The generated by a subset of a group is a normal subgroup (ie the smallest normal subgroup containing ) is nothing else than the finest equivalence relation which identifies all of the elements in each other and at the same time is compatible with the group operation ( ie is a congruence ). Just as the average of all containing a normal subgroup is the intersection of all equivalence relations is on containing and which respect the group link.
The same applies, mutatis mutandis, for the construction of ideals and corresponding congruence relations on rings.
Topologies
In the topology of systems of generators are often referred to as the base or sub-base. These are the amounts of open subsets of a topological space with the property that they generate the topology.
The latter clearly means that finite intersections and arbitrary unions can be generated every open set by the two set-theoretic operations.
Formally considered is the coarsest topology with respect to which the quantities are open. Thus on the average of all topologies containing.
σ - algebras
In the measure and integration theory, one examines the so-called σ - algebras. Consider, for example, a topological space T and examined in this a smallest σ - algebra that contains all open sets. The uniquely determined by σ - algebra is called the Borel σ - algebra. This is in the theory of integration of central importance. Here is the second form of said principle in the foreground, because the object can be specified as such is difficult to explicitly.
Amount Theoretical formulation
It should be a basic amount and a system of subsets of given. These subsets correspond to the mathematical objects that are considered in the following. In the above example of vector spaces is therefore, and the set of subspaces of. Be passed on a lot. Then asked for the smallest quantity such that applies. The amount is therefore the generating system, applies in the example above. Such an element exists and is uniquely determined, if applies
And that is true then
This applies to all of the above examples. In the case of groups, the amount of system under consideration is the set of subgroups of a group and the basic amount. In the case of σ - algebras of the system corresponds to the set of σ - algebras and the basic quantity analogous to the power set. This applies mutatis mutandis to all other examples mentioned.