algebraic system
An algebraic system, loosely speaking, is a set, together with some operations^{} 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 ${A}^{n}$ and whose range is a subset of $A$. Here, $n$ is a nonnegative 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 nonnegative 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 $\bm{A}$, 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 identity^{}, a unary operator called the multiplicative inverse, and a binary operator called the multiplication^{}.
For a more comprehensive listing of examples, please see this entry (http://planetmath.org/ExamplesOfAlgebraicSystems).
Remarks.

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An algebraic system is also called an algebra^{} for short. Some authors require that $A$ be nonempty. Note that $A$ is automatically nonempty if $O$ contains constants. A finite algebra is an algebra whose underlying set is finite.

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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 generalizations^{} 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 algebra^{}. Finally, if more than one underlying set is involved, then the algebra is said to be manysorted.
The study of algebraic systems is called the theory of universal algebra^{}. 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 onetoone 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 $\tau $, whose elements are called operator symbols, such that there is a function $a:\tau \to \mathbb{N}\cup \{0\}$. Given an operator symbol $f$, its image $a(f)$ is called the arity of $f$.
Remark. It is often the practice to wellorder $\tau $, and write $\tau $ as a sequence^{} of nonnegative integers $\u27e8a({f}_{1}),a({f}_{2}),\mathrm{\dots},\u27e9$. When $\tau $ is finite, the convention is to order the sequence in nonincreasing order: $a({f}_{1})\ge a({f}_{2})\ge \mathrm{\cdots}\ge a({f}_{n})$.
Definition. An algebraic system $\bm{A}$ is said to be of type $\tau $ if there is a bijection between $O$ and $\tau $ so that every operator symbol $f$ in $\tau $ corresponds to an operator ${f}_{\bm{A}}$ of arity $a(f)$ in $O$. When the algebra $\bm{A}$ is said to be of type $\tau $, we also say that $\bm{A}$ is a $\tau $algebra.
For example, a group is an algebraic system of type $\u27e82,1,0\u27e9$, 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.
References
 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: http://math.chapman.edu/cgibin/structures^{}?HomePageMathematical Structures: Homepage
Title  algebraic system 
Canonical name  AlgebraicSystem 
Date of creation  20130322 15:44:37 
Last modified on  20130322 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  signature^{} 
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 