Outline of algebraic structures

In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.
Another branch of mathematics known as universal algebra studies algebraic structures in general. From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used. Some axiomatic formal systems that are neither varieties nor quasivarieties, called "nonvarieties", are sometimes included among the algebraic structures by tradition.
Concrete examples of each structure will be found in the articles listed.
Algebraic structures are so numerous today that this article will inevitably be incomplete. In addition to this, there are sometimes multiple names for the same structure, and sometimes one name will be defined by disagreeing axioms by different authors. Most structures appearing on this page will be common ones which most authors agree on. Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath. These lists mention many structures not included below, and may present more information about some structures than is presented here.
Study of algebraic structures.
Algebraic structures appear in most branches of mathematics, and one can encounter them in many different ways.
Types of algebraic structures.
In full generality, an algebraic structure may use any number of sets and any number of axioms in its definition. The most commonly studied structures, however, usually involve only one or two sets and one or two binary operations. The structures below are organized by how many sets are involved, and how many binary operations are used. Increased indentation is meant to indicate a more exotic structure, and the least indented levels are the most basic.
One binary operation on one set.
The following structures consist of a set with a binary operation. The most common structure is that of a "group". Other structures involve weakening or strengthening the axioms for groups, and may additionally use unary operations.
Two binary operations on one set.
The main types of structures with one set having two binary operations are "rings" and "lattices". The axioms defining many of the other structures are modifications of the axioms for rings and lattices. One major difference between rings and lattices is that their two operations are related to each other in different ways. In ring-like structures, the two operations are linked by the distributive law; in lattice-like structures, the operations are linked by the absorption law.
Two binary operations and two sets.
The following structures have the common feature of having two sets, "A" and "B", so that there is a binary operation from "A"×"A" into "A" and another operation from "A"×"B" into "A".
Three binary operations and two sets.
Many structures here are actually hybrid structures of the previously mentioned ones.
Algebraic structures with additional non-algebraic structure.
There are many examples of mathematical structures where algebraic structure exists alongside non-algebraic structure.
Algebraic structures in different disciplines.
Some algebraic structures find uses in disciplines outside of abstract algebra. The following is meant to demonstrate some specific applications in other fields.
In Physics:
In Mathematical logic:
In Computer science:
References.
A monograph available free online: