Algebraic structure

Builds on Signature Structure Finitary operation

Definition

Let $\sigma$ be a signature whose symbols are all function symbols, each with an arity $n \ge 0$ . An algebraic structure of signature $\sigma$ , or $\sigma$ -algebra, is a pair $\mathbf{A} = (A, (f^{\mathbf{A}})_{f \in \sigma})$ such that:

$A$ is a nonempty set, the universe of $\mathbf{A}$ ;
for each symbol $f \in \sigma$ of arity $n$ , $f^{\mathbf{A}} \colon A^n \to A$ is an $n$ -ary finitary operation on $A$ .

An operation of arity $0$ is identified with an element of $A$ , a constant. So a $\sigma$ -algebra is the same as a structure for the purely functional signature $\sigma$ .

Remarks

Example: $(\Z, +, -, 0)$ is an algebraic structure of signature $(2, 1, 0)$ . Addition is a map $\Z^2 \to \Z$ , negation is a map $\Z \to \Z$ , and $0 \in \Z$ is a constant.

Non-example: $(\N, +, -, 0)$ with $-$ the map $x \mapsto -x$ is not an algebraic structure, because $-1 \notin \N$ , so clause 2 fails for the unary symbol.

A class of algebras, such as the class of groups, is given by a signature together with axioms that the operations must satisfy. The axioms are not part of the definition of an algebraic structure. Some authors allow the empty set as universe when $\sigma$ has no constants.

Reading path

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Finitary operation concept
Signature concept
Structure concept
Algebraic structure definition

Tag	`7XQBJ52RA1`
URI	`m2://7XQBJ52RA1`
Kind	definition
Writing	W0 stub
MSC	08A05, 08A62
Defines	Algebraic structure
Uses	Signature, Structure, Finitary operation
Changed	2 November 2026

Algebraic structure

Definition

Remarks

Reading path

Further reading

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