In mathematics, an algebraic structure or algebraic system[1] consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities (known as axioms) that these operations must satisfy.
Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic.
Sets with one or more operations that obey specific laws are called algebraic structures.
When a new problem involves the same laws as such an algebraic structure, all the results that have been proved using only the laws of the structure can be directly applied to the new problem.
An axiom of an algebraic structure often has the form of an identity, that is, an equation such that the two sides of the equals sign are expressions that involve operations of the algebraic structure and variables.
If the variables in the identity are replaced by arbitrary elements of the algebraic structure, the equality must remain true.
The introduction of such auxiliary operation complicates slightly the statement of an axiom, but has some advantages.
Given a specific algebraic structure, the proof that an existential axiom is satisfied consists generally of the definition of the auxiliary function, completed with straightforward verifications.
Also, when computing in an algebraic structure, one generally uses explicitly the auxiliary operations.
For example, in the case of numbers, the additive inverse is provided by the unary minus operation
Also, in universal algebra, a variety is a class of algebraic structures that share the same operations, and the same axioms, with the condition that all axioms are identities.
What precedes shows that existential axioms of the above form are accepted in the definition of a variety.
(it follows that fields do not form a variety in the sense of universal algebra.)
Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations.
The study of varieties is an important part of universal algebra.
Some structures do not form varieties, because either: Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and division rings.
Structures with nonidentities present challenges varieties do not.
Category theory is another tool for studying algebraic structures (see, for example, Mac Lane 1998).
Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms) is a category of topological spaces with extra structure.
There are various concepts in category theory that try to capture the algebraic character of a context, for instance In a slight abuse of notation, the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself.
For example, the sentence, "We have defined a ring structure on the set