Variety of algebras

Definition

Let $\sigma$ be a signature of function symbols, and let $\Sigma$ be a set of identities $s \approx t$ in the language of $\sigma$ . An algebra $A$ of signature $\sigma$ satisfies $s \approx t$ if $s$ and $t$ take the same value in $A$ for every assignment of the variables. The class $\operatorname{Mod}(\Sigma)$ is the class of all algebras of signature $\sigma$ that satisfy every identity in $\Sigma$ . A variety (or equational class) of signature $\sigma$ is a class $\mathcal{V}$ of algebras of signature $\sigma$ with $\mathcal{V} = \operatorname{Mod}(\Sigma)$ for some $\Sigma$ .

Remarks

Example: abelian groups, in the signature $(+, -, 0)$ , form a variety. It is $\operatorname{Mod}(\Sigma)$ for $\Sigma$ the identities $(x + y) + z \approx x + (y + z)$ , $x + 0 \approx x$ , $x + (-x) \approx 0$ and $x + y \approx y + x$ . Non-example: fields do not form a variety of signature $(+, \cdot, -, 0, 1)$ . Every variety is closed under direct products, but the product $\F_2 \times \F_2$ of two fields is not a field, since $(1, 0) \cdot (0, 1) = (0, 0)$ . By Birkhoff's theorem, a class is a variety if and only if it is closed under homomorphic images, subalgebras and direct products. Some authors write $\operatorname{Mod}(\Sigma)$ as $V(\Sigma)$ .

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Identity concept
Variety of algebras definition

Tag	`SARA216BGB`
URI	`m2://SARA216BGB`
Kind	definition
Writing	W1 stated
MSC	08B99, 03C05
Defines	Variety of algebras
Uses	Identity
Changed	2 November 2026

Variety of algebras

Definition

Remarks

Reading path

Further reading

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