Enclose (mathematical)

One speaks about fence or closing in mathematics in very diverse contexts. Some examples are listed below.

Enclose for operations

In Mathematical, one says that a Ensemble is closed for functions or operation S if these Opération S applied to elements of the unit produces an element of the unit. It is also said that the unit is stable for these operations. For example, the whole of the Real numbers is closed for the Soustraction, but not the whole of the natural Entiers, 3 and 7 are both of the natural entireties, but (3 − 7) is not it.

A precondition to speak about algebraic structures is that this one is closed under the operations concerned, for example one can speak about Groupe only for unit provided with a binary operation, the unit being closed for this operation.

That is to say L a list of operations. One calls fence of a unit S for the operations of L inside a unit E even closed given him for these operations, the smallest subset of E container S and closed for the operations of L . When it is indeed Opération S in a narrow direction, functions having a finished number of arguments in E , everywhere defined, and with value in E , it is checked that the intersection of all the parts of E closed containing S and closed under L defines the fence of S . There is at least a closed part for the operations of L , the unit “E” him even, and thus it is quite possible to define this intersection. The fact of being closed by the operations of L is stable by intersection, therefore the unit thus defined is well closed under the operations of L . Thus, the fence for the subtraction of the whole of the natural entireties, seen like subset of the real numbers, is the whole of the relative entireties (in the whole of realities as in any additive group whose natural entireties are a under-monoid). If only the fence is not supposed makes inside an additive group which “extends” the addition of the natural entireties and for the concept of subtraction in this additive group, it does not have there no reason which the fence gives the relative entireties.

In algebra one will speak rather about generated substructure, for example of sub-group generated by a subset. In Theory of the languages, the under-monoid of the words (provided with the operation of concatenation) generated by a language is called Fermeture of Kleene language.

In Set theory, the whole of the natural entireties is defined like the smallest unit containing 0 is closed for the operation successor (for a unit 0 and one operation successor which one will have chosen injective, and such as 0 is not the image of an element by a successor). The situation is a little different, in the theory ZFC, the universe, which is closed for the operation successor chosen, is not a unit (an object of the theory). One can define by fence the class entireties (a part, with the intuitive direction, universe). So that it is a unit, it should be supposed that there exists at least a unit containing 0 and closed by successor: it is the Axiome of infinite the. For this example one sees that to be able to speak about fence of S under certain operations, it is essential to have supposed or have shown before the existence of a unit closed under these operations and container S .

If one returns the following the example of of the natural entireties and fence by subtraction, the fact of supposing the existence of realities for contruire the whole of the relative entireties is completely artificial: one builds relative the before realities; the unit closed by subtraction (or passage on the other hand) whose one shows the existence, is directly the whole of the relative ones.

enclose binary relations

A binary Relation on a unit E is defined by its graph which is a whole of couples of elements of E . The transitive end of a relation R is the smallest relation Transitive containing this relation. That can be seen as a fence for a partial function on the couples (that is not difficult to make it total if one holds so that it is a particular case of the definite concept in the preceding paragraph). One defines in a similar way the reflexive, symmetrical fence etc of a binary relation.

Generalizations

the concept of fence described in the preceding paragraphs spreads in various ways, and can take different names according to the context.

First of all, one can close a whole by relations, and not by only by operations. For example the vectorial subspace generated by a part S of a vector Space E , perhaps described as the fence by linear Combination of S . The vectorial subspaces are the subsets of E closed by linear Combinaison. That does not concern exactly the preceding diagram, since one utilizes the product by a scalar. One can see that as the fence by the ternary relation, “the vector W is combination linear of the two vectors U and v ”. In a similar way, in Geometry, the convex Enveloppe is also a fence (by Barycentre S with plus coefficients).

In Set theory the transitive end of a unit is smallest Ensemble transitive the container, it thus acts of the fence for the relation of membership.

But the fence by relations does not exhaust either the uses of this term.

In Algèbre the algebraic Clôture of a body is the smallest body algebraically closed the container (defined except for isomorphism), whose one shows the existence directly. There still one can see this like a fence for the algebraic concept of Extension.

In Logique mathematics a theory is often defined as a whole of statements closed by deduction (inside the whole of all the statements, which is obviously closed to him by deduction). The relation of deduction can be defined like a relation between a finished whole (or a finished succession) of formulas and a formula.

In Topology closed real line, for example the closed Intervals like = { X : 1 ≤ X ≤ 2}, is closed by passage in extreme cases of the convergent continuation S (indexed by entireties) of realities of closed, and this property is characteristic. Known as more simply a subset S real line is closed if is only if any continuation of points of S convergent has as a limit a point of S . This property spreads such as it is with a metric Espace, and better with a Espace at countable base. For a topological Espace in general it is necessary to speak about filter rather than of continuation.

Rather than about fence by passage in extreme cases, one speaks about adherence, or (sometimes of closing).

Matroïde S, the closing of X is largest Surensemble of X which has the same row as X . -->

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