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| title | chunk | source | category | tags | date_saved | instance |
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| Outline of algebraic structures | 1/3 | https://en.wikipedia.org/wiki/Outline_of_algebraic_structures | reference | science, encyclopedia | 2026-05-05T08:12:44.195968+00:00 | kb-cron |
In mathematics, many types of algebraic structures 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 Archived 2007-11-13 at the Wayback Machine. 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.
Beginning study: In American universities, groups, vector spaces and fields are generally the first structures encountered in subjects such as linear algebra. They are usually introduced as sets with certain axioms. Advanced study: Abstract algebra studies properties of specific algebraic structures. Universal algebra studies algebraic structures abstractly, rather than specific types of structures. Varieties Category theory studies interrelationships between different structures, algebraic and non-algebraic. To study a non-algebraic object, it is often useful to use category theory to relate the object to an algebraic structure. Example: The fundamental group of a topological space gives information about the topological space.
== 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 set with no binary operations === Set: a degenerate algebraic structure S having no operations. Pointed set: S has one or more distinguished elements, often 0, 1, or both. Unary system: S and a single unary operation over S. Pointed unary system: a unary system with S a pointed set.
=== One binary operation on one set ===
The following group-like structures consist of a set with a binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition). 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.
Magma or groupoid: S and a single binary operation over S. Semigroup: an associative magma. Monoid: a semigroup with identity element. Group: a monoid with a unary operation (inverse), giving rise to inverse elements. Abelian group: a group whose binary operation is commutative. Quasigroup: a magma obeying the Latin square property. A quasigroup may also be represented using three binary operations. Loop: a quasigroup with identity. Semilattice: a semigroup whose operation is idempotent and commutative. The binary operation can be called either meet or join. This is basically "half" of a lattice structure (see below).
=== Two binary operations on one set === The main types of structures with one set having two binary operations are ring-like or ringoids and lattice-like or simply lattices. Ringoids and lattices can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the distributive law; in the case of lattices, they are linked by the absorption law. Ringoids also tend to have numerical models, while lattices tend to have set-theoretic models. In ring-like structures or ringoids, the two binary operations are often called addition and multiplication, with multiplication linked to addition by the distributive law.