A mathematical structure is an association between one set of mathematical objects with another in order to give those objects additional meaning or power. Common mathematical objects include *numbers*, *sets*, *relations*, and *functions*.

An **algebraic structure** is defined by a collection of objects and the operations which are allowed to be applied to those objects. A structure may consist of a single set or multiple sets of mathematical objects (e.g. numbers). These sets are **closed** under a particular operation or operations, meaning that the result of the operation applied to any element of the set is also in the set. **Axioms** are conditions which the sets and operations must satisfy.

A simple but ubiquitous algebraic structure is the **group**. A group is a set and a single binary operation, usually denoted , which satisfy the axiom of **associativity** and contain and **identity** element and **inverse** elements. Associativity specifies that the order in which the operations are performed does not matter; that is . The identity element is a special element such that the operation applied to it and another element results in the other element; formally: . Each element of the set must have an inverse element that yields the identity element when the two are combined: . If the axiom of **commutativity** is added, the group is referred to as an **Abelian group**. Commutativity allows the operands to be reorganized: . If the requirement of inverse elements is removed from the group structure, the structure is called a **monoid**.

A **group homomorphism** is a function which preserves the relationships between the elements of the set, or the *group structure*. For groups and , is a homomorphism iff . If the map is *invertible*, i.e. it has an inverse such that , then is said to be an **isomorphism***. A ***group endomorphism** is a homomorphism from a group onto itself, , while an invertible endomorphism is called a **automorphism**.* *A **subgroup** is a group within a larger group. For a subgroup of a group , and the identity element of is the identity element of .

A **ring** is an algebraic structure which adds another operations and a few axioms to the group structure. The operations and of a ring satisfy different axioms. The set and the *addition* operator must form an *Abelian group* as described above, while the set and the *multiplication* operator must form a *monoid*. In addition, the operators must satisfy the axiom of **distribution**, specifically the operator must be able to distribute over the operator, and . If forms a *commutative monoid*, that is, a monoid with commutativity, then the ring is said to be a **commutative ring**.

Similarly to a group, a ring may also have a **ring homomorphism** if satisfies and . Likewise, is an *isomorphism* if it has an inverse that satisfies the identity relation described for group isomorphism above. A ring is a **subring** of a ring if and contains the multiplicative identity from .

An algebraic structure is a **field** if it satisfies the axioms of a *ring* with a few addition axioms. Both operators of a field must satisfy *commutativity*, and the set must contain inverses under both operators, except that the field may not contain a multiplicative inverse for the additive identity element. Another way to describe a field is to say that the *additive group* is an Abelian group, and the *multiplicative group* without the additive identity is also an Abelian group. The inclusion of inverses for both operators lead to the intuitive notion of *subtraction* and *division* (except division by 0). An example of a well known field is the field of real numbers, .

A **metric space** is a mathematical structure , where is a binary function which defines the real-valued *distance* between two elements of the set . A distance is a non-negative quantity, and only equal to zero when the two element are equal. A distance should also satisfy the axiom of **symmetry**, and the **triangle inequality** . For example, the real-valued vector space equipped with the Euclidean distance metric yields the Euclidean metric space.

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