Varieties of universal algebras

A \emph{variety} is a class of structures of the same signature that is defined by a set of identities, i.e., universally quantified equations or, more generally, atomic formulas.

Varieties are also called \emph{equational classes}.

By a fundamental result of Birkhoff1) a class $\mathcal{K}$ of algebras is a variety iff it is closed under the operators $H$, $S$, $P$ (i.e., $H\mathcal{K}\subseteq\mathcal{K}$, $S\mathcal{K}\subseteq\mathcal{K}$, and $P\mathcal{K}\subseteq\mathcal{K}$), where

$H\mathcal{K}=\{$homomorphic images of members of $\mathcal{K}\}$
$S\mathcal{K}=\{$subalgebras of members of $\mathcal{K}\}$
$P\mathcal{K}=\{$direct products of members of $\mathcal{K}\}$.

Equivalently, $\mathcal K$ is a variety iff $\mathcal K=HSP\mathcal K$.

In particular, given any class $\mathcal K$ of algebras, $V\mathcal K=HSP\mathcal K$ is the smallest variety that contains $\mathcal K$, and is called the \emph{variety generated by $\mathcal K$}.

See Stanley N. Burris and H.P. Sankappanavar, A Course in Universal Algebra for more details.

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A picture of some theories ordered by interpretability

Some varieties and quasivarieties listed by signature and (first) subclass relation

Proper quasivarieties are marked by a *

$\langle \rangle$ Sets

$\langle 0\rangle$ Pointed sets

$\langle 1\rangle$ Mono-unary algebras

$\langle 1,0\rangle$ Pointed mono-unary algebras

$\langle 1,1\rangle$ Duo-unary algebras

$\langle 1,1,\ldots\rangle$ Unary algebras

$\langle 2\rangle$ Groupoids

$\langle 2,0\rangle$ Pointed groupoids

$\langle 2,1\rangle$ Groupoids with a unary operation

$\langle 2,1,0\rangle$ Pointed groupoids with a unary operation

$\langle 2,1,0,1,1,\ldots\rangle$ Pointed groupoids with a unary operations

$\langle 2,2\rangle$ Duo-groupoids

$\langle 2,2,0\rangle$ Pointed duo-groupoids

$\langle 2,2,1\rangle$

$\langle 2,2,\ldots\rangle$

$\langle 2,0,2,0\rangle$

$\langle 2,1,0,2\rangle$

$\langle 2,1,0,2,0\rangle$

$\langle 2,0,2,0,1\rangle$

$\langle 2,0,2,0,1,1\rangle$

$\langle 2,0,2,0,1,1\rangle$

$\langle 2,0,2,0,1,2\rangle$

$\langle 2,0,2,0,1,2,0\rangle$

$\langle 2,0,2,0,1,2,1,0\rangle$

$\langle 2,0,2,0,1,2,0,2,2\rangle$

$\langle 2,0,2,0,1,\ldots\rangle$

$\langle 2,0,2,0,\ldots\rangle$

$\langle 2,0,2,0,\ldots\rangle$

$\langle 2,2,2\rangle$

$\langle 2,2,2,0\rangle$

$\langle 2,2,2,1,0\rangle$

$\langle 2,2,2,0,2\rangle$

$\langle 2,2,2,0,2,2\rangle$

$\langle 2,0,2,0,2,2\rangle$

$\langle 2,0,2,0,2,0,2\rangle$

$\langle 2,0,2,0,2,0,2,2\rangle$

$\langle 2,0,2,0,1,2,2\rangle$

$\langle 2,2,0,2,0,1,2,2\rangle$

1) Garrett Birkhoff, \emph{On the structure of abstract algebras}, Proceedings of the Cambridge Philosophical Society, 31:433–454, 1935

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