Varieties of universal algebras
A
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
equational classes.
By a fundamental result of [Garrett Birkhoff,
On the structure of abstract algebras,
Proceedings of the Cambridge Philosophical Society, 31:433--454, 1935] 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 $\mathcal{K}\}$.
See
Stanley N. Burris and H.P. Sankappanavar, A Course in Universal Algebra for
more details.
Show all pages on
varieties
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 (LaTeX)
$\langle 0\rangle$
Pointed sets (LaTeX)
$\langle 1\rangle$
Mono-unary algebras (LaTeX)
$\langle 1,0\rangle$
Pointed mono-unary algebras (LaTeX)
$\langle 1,1\rangle$
Duo-unary algebras (LaTeX)
$\langle 1,1,\ldots\rangle$
Unary algebras (LaTeX)
$\langle 2\rangle$
Groupoids (LaTeX)
$\langle 2,0\rangle$
Pointed groupoids (LaTeX)
$\langle 2,1\rangle$
Groupoids with a unary operation (LaTeX)
$\langle 2,1,0\rangle$
Pointed groupoids with a unary operation (LaTeX)
$\langle 2,1,0,1,1,\ldots\rangle$
Pointed groupoids with a unary operations (LaTeX)
$\langle 2,2\rangle$
Duo-groupoids (LaTeX)
$\langle 2,2,0\rangle$
Pointed duo-groupoids (LaTeX)
$\langle 2,2,1\rangle$
$\langle 2,2,\ldots\rangle$
$\langle 2,0,2,0\rangle$
$\langle 2,1,0,2\rangle$
$\langle 2,1,0,2,0\rangle$
$\langle 2,0,2,0,1\rangle$
$\langle 2,0,2,0,1,1\rangle$
$\langle 2,0,2,0,1,1\rangle$
$\langle 2,0,2,0,1,2\rangle$
$\langle 2,0,2,0,1,2,0\rangle$
$\langle 2,0,2,0,1,2,1,0\rangle$
$\langle 2,0,2,0,1,2,0,2,2\rangle$
$\langle 2,0,2,0,1,\ldots\rangle$
$\langle 2,0,2,0,\ldots\rangle$
$\langle 2,0,2,0,\ldots\rangle$
$\langle 2,2,2\rangle$
$\langle 2,2,2,0\rangle$
$\langle 2,2,2,1,0\rangle$
$\langle 2,2,2,0,2\rangle$
$\langle 2,2,2,0,2,2\rangle$
$\langle 2,0,2,0,2,2\rangle$
$\langle 2,0,2,0,2,0,2\rangle$
$\langle 2,0,2,0,2,0,2,2\rangle$
$\langle 2,0,2,0,1,2,2\rangle$
$\langle 2,2,0,2,0,1,2,2\rangle$
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