algebraic system

An algebraic system, loosely speaking, is a set, together with some operationsMathworldPlanetmath on the set. Before formally defining what an algebraic system is, let us recall that a n-ary operation (or operator) on a set A is a function whose domain is An and whose range is a subset of A. Here, n is a non-negative integer. When n=0, the operation is usually called a nullary operation, or a constant, since one element of A is singled out to be the (sole) value of this operation. A finitary operation on A is just an n-ary operation for some non-negative integer n.

Definition. An algebraic system is an ordered pair (A,O), where A is a set, called the underlying set of the algebraic system, and O is a set, called the operator set, of finitary operations on A.

We usually write 𝑨, instead of (A,O), for brevity.

A prototypical example of an algebraic system is a group, which consists of the underlying set G, and a set O consisting of three operators: a constant e called the multiplicative identityPlanetmathPlanetmath, a unary operator called the multiplicative inverse, and a binary operator called the multiplicationPlanetmathPlanetmath.

For a more comprehensive listing of examples, please see this entry (


  • An algebraic system is also called an algebraMathworldPlanetmathPlanetmath for short. Some authors require that A be non-empty. Note that A is automatically non-empty if O contains constants. A finite algebra is an algebra whose underlying set is finite.

  • By definition, all operators in an algebraic system are finitary. If we allow O to contain infinitary operations, we have an infinitary algebraic system. Other generalizationsPlanetmathPlanetmath are possible. For example, if the operations are allowed to be multivalued, the algebra is said to be a multialgebra. If the operations are not everywhere defined, we get a partial algebraMathworldPlanetmath. Finally, if more than one underlying set is involved, then the algebra is said to be many-sorted.

The study of algebraic systems is called the theory of universal algebraMathworldPlanetmath. The first important thing in studying algebraic system is to compare systems that are of the same ‘‘type’’. Two algebras are said to have the same type if there is a one-to-one correspondence between their operator sets such that an n-ary operator in one algebra is mapped to an n-ary operator in the other algebra. A more formal way of doing this is to define what a type is:

Definition. A type is a set τ, whose elements are called operator symbols, such that there is a function a:τ{0}. Given an operator symbol f, its image a(f) is called the arity of f.

Remark. It is often the practice to well-order τ, and write τ as a sequenceMathworldPlanetmath of non-negative integers a(f1),a(f2),,. When τ is finite, the convention is to order the sequence in non-increasing order: a(f1)a(f2)a(fn).

Definition. An algebraic system 𝑨 is said to be of type τ if there is a bijection between O and τ so that every operator symbol f in τ corresponds to an operator f𝑨 of arity a(f) in O. When the algebra 𝑨 is said to be of type τ, we also say that 𝑨 is a τ-algebra.

For example, a group is an algebraic system of type 2,1,0, where 2 is the arity of the group multiplication, 1 is the arity of the group inverse, and 0 is the arity of the group multiplicative identity.


  • 1 А. И. Мальцев: Алгебраические  системы.  Издательство  ‘‘Наука’’. Москва (1970).
  • 2 P. M. Cohn: Universal Algebra, Harper & Row, (1965).
  • 3 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
  • 4 P. Jipsen: Structures: Homepage
Title algebraic system
Canonical name AlgebraicSystem
Date of creation 2013-03-22 15:44:37
Last modified on 2013-03-22 15:44:37
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 50
Author CWoo (3771)
Entry type Definition
Classification msc 08A05
Classification msc 03E99
Classification msc 08A62
Synonym algebraic structure
Synonym universal algebra
Synonym signaturePlanetmathPlanetmath
Synonym trivial algebra
Related topic RelationalSystem
Related topic Model
Related topic StructuresAndSatisfaction
Related topic PartiallyOrderedAlgebraicSystem
Defines n-ary operator
Defines finitary operator
Defines infinitary operator
Defines operator set
Defines constant operator
Defines operator symbol
Defines nullary operator
Defines type
Defines trivial algebraic system
Defines finite algebra