linguistic:voice_change
In algebraic structures

An algebraic structure is a set equipped with operations that satisfy some axioms. These axioms may be identities. Some axioms may contain existential quantifiers ∃; {\displaystyle \exists ;} in this case it is worth to add some auxiliary operations in order that all axioms become identities or purely universally quantified formulas. See Algebraic structure for details. 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.

For example, a group is a set with an associative operation, often called multiplication, with an identity element, such that every element has an inverse element. Here, the auxiliary operations are the nullary operation that results in the identity element and the unary operation of inversion. 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. So, a non-empty subset of a group that is closed under multiplication and inversion is a group that is called a subgroup. The subgroup generated by a single element, that is, the closure of this element, is called a cyclic group.

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.