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 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 $\boldsymbol{A}$ , 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.

Remarks.

• An algebraic system is also called an algebra 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 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 $n$ -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 $a:\tau \to \mathbb{N}\cup \lbrace 0\rbrace$ . Given an operator symbol $f$ , its image $a(f)$ is called the arity of $f$ .

Remark. It is often the practice to well-order $\tau$ , and write $\tau$ as a sequence of non-negative integers $\langle a(f_1), a(f_2), \ldots, \rangle$ . When $\tau$ is finite, the convention is to order the sequence in non-increasing order: $a(f_1)\ge a(f_2)\ge \cdots \ge a(f_n)$ .

Definition. An algebraic system $\boldsymbol{A}$ 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 $f_{\boldsymbol{A}}$ of arity $a(f)$ in $O$ . When the algebra $\boldsymbol{A}$ is said to be of type $\tau$ , we also say that $\boldsymbol{A}$ is a $\tau$ -algebra.

For example, a group is an algebraic system of type $\langle 2,1,0\rangle$ , 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.

Bibliography

1

. . :. ``''. (1970).

2

P. M. Cohn: Universal Algebra, Harper & Row, (1965).

3

G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).

4

P. Jipsen: Mathematical Structures: Homepage