Algebraic structure

related topics
{math, number, function}
{group, member, jewish}
{math, energy, light}
{language, word, form}

In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic structure has been formalized in universal algebra.

As an abstraction, an "algebraic structure" is the collection of all possible models of a given set of axioms. More concretely, an algebraic structure is any particular model of some set of axioms. For example, the monster group both "is" an algebraic structure in the concrete sense, and abstractly, "has" the group structure in common with all other groups. This article employs both meanings of "structure."

This definition of an algebraic structure should not be taken as restrictive. Anything that satisfies the axioms defining a structure is an instance of that structure, regardless of how many other axioms that instance happens to have. For example, all groups are also semigroups and magmas.

Contents

Structures whose axioms are all identities

If the axioms defining a structure are all identities, the structure is a variety (not to be confused with algebraic variety in the sense of algebraic geometry). Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. 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.

All structures in this section are varieties. Some of these structures are most naturally axiomatized using one or more nonidentities, but are nevertheless varieties because there exists an equivalent axiomatization, one perhaps less perspicuous, composed solely of identities. Algebraic structures that are not varieties are described in the following section, and differ from varieties in their metamathematical properties.

Full article ▸

related documents
Linear combination
E (mathematical constant)
Ideal class group
Sequence
Elliptic curve
IEEE 754-1985
Abstract interpretation
Gaussian quadrature
Random variable
Natural transformation
Binary relation
Absolute convergence
Galois theory
Partition (number theory)
Pushdown automaton
Ultrafilter
Axiom schema of replacement
Presentation of a group
Complex analysis
Heine–Borel theorem
Fuzzy logic
Euclidean algorithm
Glossary of topology
Database normalization
Exclusive or
Countable set
Factorization
Linear independence
Type theory
Filter (mathematics)