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 -ary operation (or operator) on a set is a function whose domain is and whose range is a subset of . Here, is a non-negative integer. When , the operation is usually called a nullary operation, or a constant, since one element of is singled out to be the (sole) value of this operation. A finitary operation on is just an -ary operation for some non-negative integer .
Definition. An algebraic system is an ordered pair , where is a set, called the underlying set of the algebraic system, and is a set, called the operator set, of finitary operations on .
We usually write , instead of , for brevity.
A prototypical example of an algebraic system is a group, which consists of the underlying set , and a set consisting of three operators: a constant 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).
An algebraic system is also called an algebra for short. Some authors require that be non-empty. Note that is automatically non-empty if 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 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 many-sorted.
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 one-to-one correspondence between their operator sets such that an -ary operator in one algebra is mapped to an -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 , whose elements are called operator symbols, such that there is a function . Given an operator symbol , its image is called the arity of .
Remark. It is often the practice to well-order , and write as a sequence of non-negative integers . When is finite, the convention is to order the sequence in non-increasing order: .
Definition. An algebraic system is said to be of type if there is a bijection between and so that every operator symbol in corresponds to an operator of arity in . When the algebra is said to be of type , we also say that is a -algebra.
For example, a group is an algebraic system of type , where is the arity of the group multiplication, is the arity of the group inverse, and 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: http://math.chapman.edu/cgi-bin/structures?HomePageMathematical Structures: Homepage
|Date of creation||2013-03-22 15:44:37|
|Last modified on||2013-03-22 15:44:37|
|Last modified by||CWoo (3771)|
|Defines||trivial algebraic system|