Depending on the context, X is called the closure of Y or the set generated or spanned by Y.
in this case it is worth to add some auxiliary operations in order that all axioms become identities or purely universally quantified formulas.
A set with a single binary operation that is closed is called a magma.
In this context, given an algebraic structure S, a substructure of S is a subset that is closed under all operations of S, including the auxiliary operations that are needed for avoiding existential quantifiers.
A substructure is an algebraic structure of the same type as S. It follows that, in a specific example, when closeness is proved, there is no need to check the axioms for proving that a substructure is a structure of the same type.
Given a subset X of an algebraic structure S, the closure of X is the smallest substructure of S that is closed under all operations of S. In the context of algebraic structures, this closure is generally called the substructure generated or spanned by X, and one says that X is a generating set of the substructure.
A subset of a group that is closed under multiplication and inversion is also closed under the nullary operation (that is, it contains the identity) if and only if it is non-empty.
In linear algebra, the closure of a non-empty subset of a vector space (under vector-space operations, that is, addition and scalar multiplication) is the linear span of this subset.
It is a vector space by the preceding general result, and it can be proved easily that is the set of linear combinations of elements of the subset.
Similar examples can be given for almost every algebraic structures, with, sometimes some specific terminology.
For example, in a commutative ring, the closure of a single element under ideal operations is called a principal ideal.
A binary relation on a set A can be defined as a subset R of
Many properties or operations on relations can be used to define closures.
Some of the most common ones follow: A preorder is a relation that is reflective and transitive.
It follows that the reflexive transitive closure of a relation is the smallest preorder containing it.
Similarly, the reflexive transitive symmetric closure or equivalence closure of a relation is the smallest equivalence relation that contains it.
In the preceding sections, closures are considered for subsets of a given set.
Given a poset S whose partial order is denoted with ≤, a closure operator on S is a function
By idempotency, an element is closed if and only if it is the closure of some element of S. An example is the topological closure operator; in Kuratowski's characterization, axioms K2, K3, K4' correspond to the above defining properties.
An example not operating on subsets is the ceiling function, which maps every real number x to the smallest integer that is not smaller than x.
This equivalence remains true for partially ordered sets with the greatest-lower-bound property, if one replace "closed sets" by "closed elements" and "intersection" by "greatest lower bound".