Outline of algebraic structures

The following outline is provided as an overview of and topical guide to 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.

• 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 binary operation on one set

Group-like structures
	Totality^α	Associativity	Identity	Invertibility	Commutativity
Semicategory	Unneeded	Required	Unneeded	Unneeded	Unneeded
Category	Unneeded	Required	Required	Unneeded	Unneeded
Groupoid	Unneeded	Required	Required	Required	Unneeded
Magma	Required	Unneeded	Unneeded	Unneeded	Unneeded
Quasigroup	Required	Unneeded	Unneeded	Required	Unneeded
Loop	Required	Unneeded	Required	Required	Unneeded
Semigroup	Required	Required	Unneeded	Unneeded	Unneeded
Monoid	Required	Required	Required	Unneeded	Unneeded
Group	Required	Required	Required	Required	Unneeded
Abelian group	Required	Required	Required	Required	Required
^α Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.

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.

• Groups are key structures. Abelian groups are an important special type of group.
  • semigroups and monoids: These are like groups, except the operation need not have inverse elements.
  • quasigroups and loops: These are like groups, except the operation need not be associative.
  • Magmas: These are like groups, except the operation need not be associative or have inverse elements.
• Semilattice: 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 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.

• Rings: The two operations are usually called addition and multiplication. Commutative rings are an especially important type of ring where the multiplication operation is commutative. Integral domains and fields are especially important types of commutative rings.
  • Nonassociative rings: These are like rings, but the multiplication operation need not be associative.
    • Lie rings and Jordan rings are special examples of nonassociative rings.
  • semirings: These are like rings, but the addition operation need not have inverses.
  • nearrings: These are like rings, but the addition operation need not be commutative.
  • *-rings: These are rings with an additional unary operation known as an involution.
• Lattices: The two operations are usually called meet and join.
  • Latticoid: meet and join commute but need not associate.
  • Skew lattice: meet and join associate but need not commute.

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.

• Vector spaces: The set A is an abelian group, and the set B is a field.
  • Graded vector spaces: Vector spaces which are equipped with a direct sum decomposition into subspaces.
• Modules: The set A is an abelian group, but the B is only a general ring and not necessarily a field.
  • Special types of modules, including free modules, projective modules, injective modules and flat modules are studied in abstract algebra.
• Group with operators: In this case, the set A is a group, and the set B is just a set.

Three binary operations and two sets

Many structures here are actually hybrid structures of the previously mentioned ones.

• Algebra over a field: This is a ring which is also a vector space over a field. There are axioms governing the interaction of the two structures. Multiplication is usually assumed to be associative.
  • Algebra over a ring: These are defined the same way as algebras over fields, except that the field may now be any commutative ring.
  • Graded algebra: These algebras are equipped with a decomposition into grades.
• Non-associative algebras: These are algebras for which the associativity of ring multiplication is relaxed.
  • Lie algebras and Jordan algebras are special examples of non-associative algebras.
• Coalgebra: This structure has axioms which make its multiplication dual to those of an associative algebra.
  • Bialgebra: These structures are simultaneously algebras and coalgebras whose operations are compatible. There are actually four operations for this structure.

Algebraic structures with additional non-algebraic structure

There are many examples of mathematical structures where algebraic structure exists alongside non-algebraic structure.

• Topological vector spaces are vector spaces with a compatible topology.
• Lie groups: These are topological manifolds that also carry a compatible group structure.
• Ordered groups, ordered rings and ordered fields have algebraic structure compatible with an order on the set.
• Von Neumann algebras: these are *-algebras on a Hilbert space which are equipped with the weak operator topology.

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:

• Lie groups are used extensively in physics. A few well-known ones include the orthogonal groups and the unitary groups.
• Lie algebras
• Inner product spaces
• Kac–Moody algebra
• The quaternions and more generally geometric algebras

In Mathematical logic:

• Boolean algebras are both rings and lattices, under their two operations.
  • Heyting algebras are a special example of boolean algebras.
• Peano arithmetic
• Boundary algebra
• MV-algebra

In Computer science:

• Max-plus algebra
• Syntactic monoid
• Transition monoid

See also

• Abstract algebra
  • Outline of abstract algebra
• Universal algebra
  • Variety (universal algebra)
• Linear algebra
  • Outline of linear algebra
• Arity
• Category theory
• Free object
• Operation (mathematics)
• Signature (logic)
• First-order theories
• Mathematical lists

Notes

References

• Garrett Birkhoff, 1967. Lattice Theory, 3rd ed, AMS Colloquium Publications Vol. 25. American Mathematical Society.
• ———, and Saunders MacLane, 1999 (1967). Algebra, 2nd ed. New York: Chelsea.
• George Boolos and Richard Jeffrey, 1980. Computability and Logic, 2nd ed. Cambridge Univ. Press.
• Dummit, David S., and Foote, Richard M., 2004. Abstract Algebra, 3rd ed. John Wiley and Sons.
• Grätzer, George, 1978. Universal Algebra, 2nd ed. Springer.
• David K. Lewis, 1991. Part of Classes. Blackwell.
• Michel, Anthony N., and Herget, Charles J., 1993 (1981). Applied Algebra and Functional Analysis. Dover.
• Potter, Michael, 2004. Set Theory and its Philosophy, 2nd ed. Oxford Univ. Press.
• Smorynski, Craig, 1991. Logical Number Theory I. Springer-Verlag.

A monograph available free online:

• Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.

External links

• Jipsen:
  • Alphabetical list of algebra structures; includes many not mentioned here.
  • Online books and lecture notes.
  • Map containing about 50 structures, some of which do not appear above. Likewise, most of the structures above are absent from this map.
• PlanetMath topic index.
• Hazewinkel, Michiel (2001) Encyclopaedia of Mathematics. Springer-Verlag.
• Mathworld page on abstract algebra.
• Stanford Encyclopedia of Philosophy: Algebra by Vaughan Pratt.