Closure (algebraic structure)

In mathematics , in particular algebra , the isolation of a set with respect to a link is understood to mean that the link between any elements of this set results in an element of the set. For example, the set of integers is closed in terms of addition , subtraction and multiplication , but not in terms of division . In the case of algebraic structures with several links, one considers the closeness of all these links accordingly.

definition

Let be a -place inner connection on a set , that is, be a function . A nonempty subset is now called closed with respect to if ${\ displaystyle f}$${\ displaystyle n}$${\ displaystyle A}$${\ displaystyle f}$${\ displaystyle A ^ {n} \ to A}$ ${\ displaystyle M \ subseteq A}$${\ displaystyle f}$

${\ displaystyle f (a_ {1}, \ dotsc, a_ {n}) \ in M}$

applies to all . That means, restricted to the definition area , a -digit inner link must also be on . ${\ displaystyle a_ {1}, \ dotsc, a_ {n} \ in M}$${\ displaystyle f}$ ${\ displaystyle M ^ {n}}$${\ displaystyle n}$${\ displaystyle M}$

Examples

• A subgroup is a non-empty subset of a group that is closed in terms of linkage and inverse formation.${\ displaystyle (G, +)}$${\ displaystyle +}$
• A sub-vector space is a non-empty subset of a vector space that is closed with regard to vector addition and scalar multiplication .${\ displaystyle V}$
• In general, an algebraic substructure is a (non-empty) subset of an algebraic structure that is closed with regard to all the links in this structure.

The importance of closeness to a link is best understood by looking at examples where it is violated.

• As a sub-structure, the group is not closed, i.e. not a sub-group. This subset is closed with regard to the addition, but not with regard to the formation of the inverse: with does not belong .${\ displaystyle (\ mathbb {N}, +)}$${\ displaystyle (\ mathbb {Z}, +, 0, -)}$${\ displaystyle a \ in \ mathbb {N}}$${\ displaystyle -a}$${\ displaystyle \ mathbb {N}}$
• The intersection of two sub-vector spaces of a vector space is always itself a sub-vector space, but the union of two sub-vector spaces is not necessarily a sub-vector space. The union is complete with regard to the scalar multiplication, but not necessarily with regard to the vector addition.

generalization

Similarly, is a subset also completed over a -digit inner join on if their image is. ${\ displaystyle M}$${\ displaystyle \ infty}$${\ displaystyle f}$${\ displaystyle A}$${\ displaystyle M}$

Example:

• If the power set of an infinite set and the set of all closed sets with respect to a T 1 topology is on , i.e. contains all (infinitely many) one-element subsets of , then a closed set with respect to the set- theoretic average is on .${\ displaystyle {\ mathcal {P}} (X)}$ ${\ displaystyle X}$${\ displaystyle {\ mathcal {C}}}$${\ displaystyle X}$${\ displaystyle {\ mathcal {C}} \ subseteq {\ mathcal {P}} (X)}$${\ displaystyle X}$${\ displaystyle {\ mathcal {C}}}$ ${\ displaystyle \ bigcap}$${\ displaystyle {\ mathcal {P}} (X)}$

The property that a link on a set always delivers uniquely certain values ​​in is also referred to as the well-defined nature of this link. ${\ displaystyle f}$${\ displaystyle A}$${\ displaystyle A}$