Closure (mathematics)

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation. For example, the positive integers are closed under addition, but not under subtraction: $1-2$ is not a positive integer even though both 1 and 2 are positive integers. Another example is the set containing only zero, which is closed under addition, subtraction and multiplication (because $0+0=0$ , $0-0=0$ , and $0\times {0}=0$ ).

Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operations individually.

Basic properties

A set that is closed under an operation or collection of operations is said to satisfy a closure property. Often a closure property is introduced as an axiom, which is then usually called the axiom of closure. Modern set-theoretic definitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous; however in practice operations are often defined initially on a superset of the set in question and a closure proof is required to establish that the operation applied to pairs from that set only produces members of that set. For example, the set of even integers is closed under addition, but the set of odd integers is not.

When a set S is not closed under some operations, one can usually find the smallest set containing S that is closed. This smallest closed set is called the closure of S (with respect to these operations). For example, the closure under subtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers. An important example is that of topological closure. The notion of closure is generalized by Galois connection, and further by monads.

The set S must be a subset of a closed set in order for the closure operator to be defined. In the preceding example, it is important that the reals are closed under subtraction; in the domain of the natural numbers subtraction is not always defined.

The two uses of the word "closure" should not be confused. The former usage refers to the property of being closed, and the latter refers to the smallest closed set containing one that may not be closed. In short, the closure of a set satisfies a closure property.

Closed sets

A set is closed under an operation if that operation returns a member of the set when evaluated on members of the set. Sometimes the requirement that the operation be valued in a set is explicitly stated, in which case it is known as the axiom of closure. For example, one may define a group as a set with a binary product operator obeying several axioms, including an axiom that the product of any two elements of the group is again an element. However the modern definition of an operation makes this axiom superfluous; an n-ary operation on S is just a subset of Sⁿ⁺¹. By its very definition, an operator on a set cannot have values outside the set.

Nevertheless, the closure property of an operator on a set still has some utility. Closure on a set does not necessarily imply closure on all subsets. Thus a subgroup of a group is a subset on which the binary product and the unary operation of inversion satisfy the closure axiom.

An operation of a different sort is that of finding the limit points of a subset of a topological space (if the space is first-countable, it suffices to restrict consideration to the limits of sequences but in general one must consider at least limits of nets). A set that is closed under this operation is usually just referred to as a closed set in the context of topology. Without any further qualification, the phrase usually means closed in this sense. Closed intervals like [1,2] = {x : 1 ≤ x ≤ 2} are closed in this sense.

A partially ordered set is downward closed (and also called a lower set) if for every element of the set all smaller elements are also in it; this applies for example for the real intervals (−∞, p) and (−∞, p], and for an ordinal number p represented as interval [ 0, p); every downward closed set of ordinal numbers is itself an ordinal number.

Upward closed and upper set are defined similarly.

P closures of binary relations

The notion of a closure can be applied for an arbitrary binary relation R ⊆ S×S, and an arbitrary property P in the following way: the P closure of R is the least relation Q ⊆ S×S that contains R (i.e. R ⊆ Q) and for which property P holds (i.e. P(Q) is true). For instance, one can define the symmetric closure as the least symmetric relation containing R. This generalization is often encountered in the theory of rewriting systems, where one often uses more "wordy" notions such as the reflexive transitive closure R^*—the smallest preorder containing R, or the reflexive transitive symmetric closure R^≡—the smallest equivalence relation containing R, and therefore also known as the equivalence closure. When considering a particular term algebra, an equivalence relation that is compatible with all operations of the algebra ^{[note 1]} is called a congruence relation. The congruence closure of R is defined as the smallest congruence relation containing R.

For arbitrary P and R, the P closure of R need not exist. In the above examples, these exist because reflexivity, transitivity and symmetry are closed under arbitrary intersections. In such cases, the P closure can be directly defined as the intersection of all sets with property P containing R.^[1]

Some important particular closures can be constructively obtained as follows:

cl_ref(R) = R ∪ { ⟨x,x⟩ : x ∈ S } is the reflexive closure of R,
cl_sym(R) = R ∪ { ⟨y,x⟩ : ⟨x,y⟩ ∈ R } is its symmetry closure,
cl_trn(R) = R ∪ { ⟨x₁,x_n⟩ : n >1 ∧ ⟨x₁,x₂⟩, ..., ⟨x_n-1,x_n⟩ ∈ R } is its transitive closure,
cl_emb,Σ(R) = R ∪ { ⟨f(x₁,…,x_i-1,x_i,x_i+1,…,x_n), f(x₁,…,x_i-1,y,x_i+1,…,x_n)⟩ : ⟨x_i,y⟩ ∈ R ∧ f ∈ Σ n-ary ∧ 1 ≤ i ≤ n ∧ x₁,...,x_n ∈ S } is its embedding closure with respect to a given set Σ of operations on S, each with a fixed arity.

The relation R is said to have closure under some cl_xxx, if R = cl_xxx(R); for example R is called symmetric if R = cl_sym(R).

Any of these four closures preserves symmetry, i.e., if R is symmetric, so is any cl_xxx(R). ^{[note 2]} Similarly, all four preserve reflexivity. Moreover, cl_trn preserves closure under cl_emb,Σ for arbitrary Σ. As a consequence, the equivalence closure of an arbitrary binary relation R can be obtained as cl_trn(cl_sym(cl_ref(R))), and the congruence closure with respect to some Σ can be obtained as cl_trn(cl_emb,Σ(cl_sym(cl_ref(R)))). In the latter case, the nesting order does matter; e.g. if S is the set of terms over Σ = { a, b, c, f } and R = { ⟨a,b⟩, ⟨f(b),c⟩ }, then the pair ⟨f(a),c⟩ is contained in the congruence closure cl_trn(cl_emb,Σ(cl_sym(cl_ref(R)))) of R, but not in the relation cl_emb,Σ(cl_trn(cl_sym(cl_ref(R)))).

Closure operator

Given an operation on a set X, one can define the closure C(S) of a subset S of X to be the smallest subset closed under that operation that contains S as a subset, if any such subsets exist. Consequently, C(S) is the intersection of all closed sets containing S. For example, the closure of a subset of a group is the subgroup generated by that set.

The closure of sets with respect to some operation defines a closure operator on the subsets of X. The closed sets can be determined from the closure operator; a set is closed if it is equal to its own closure. Typical structural properties of all closure operations are: ^[2]

The closure is increasing or extensive: the closure of an object contains the object.
The closure is idempotent: the closure of the closure equals the closure.
The closure is monotone, that is, if X is contained in Y, then also C(X) is contained in C(Y).

An object that is its own closure is called closed. By idempotency, an object is closed if and only if it is the closure of some object.

These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set.

If X is contained in a set closed under the operation then every subset of X has a closure.

Examples

In topology and related branches, the relevant operation is taking limits. The topological closure of a set is the corresponding closure operator. The Kuratowski closure axioms characterize this operator.
In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X and is closed under the operation of linear combination. This subset is a subspace.
In matroid theory, the closure of X is the largest superset of X that has the same rank as X.
In set theory, the transitive closure of a set.
In set theory, the transitive closure of a binary relation.
In algebra, the algebraic closure of a field.
In commutative algebra, closure operations for ideals, as integral closure and tight closure.
In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset.
In formal languages, the Kleene closure of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language.
In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normal subgroup containing the set.
In mathematical analysis and in probability theory, the closure of a collection of subsets of X under countably many set operations is called the σ-algebra generated by the collection.

Notes

↑ that is, such that e.g. xRy implies f(x,x₂) R f(y,x₂) and f(x₁,x) R f(x₁,y) for any binary operation f and arbitrary x₁,x₂ ∈ S
↑ formally: if R = cl_sym(R), then cl_xxx(R) = cl_sym(cl_xxx(R))

References

↑ Baader, Franz; Nipkow, Tobias (1998). Term Rewriting and All That. Cambridge University Press. pp. 8–9.
↑ Birkhoff, Garrett (1967). Lattice Theory. Colloquium Publications. 25. Am. Math. Soc. p. 111.

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