=== 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 Birkhoff[(Garrett Birkhoff, \emph{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 $\mathcal{K}\}$.

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 [[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html| Stanley N. Burris and H.P. Sankappanavar,  A Course in Universal Algebra]] for
more details.

Show all pages on [[http://math.chapman.edu/~jipsen/structures/doku.php/?do=search&id=variety&fulltext=Search| varieties]]

A picture of some [[http://www.chapman.edu/~jipsen/PCP/theoriesPO1.html| 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]]
  *[[M-sets]]
    *[[G-sets]]

$\langle 2\rangle$ [[Groupoids]]
  *[[Semigroups]]
    *[[Commutative semigroups]]
      *[[Semilattices]]
    *[[Bands]]
      *[[Normal bands]]
        *[[Rectangular bands]]
  *[[Commutative groupoids]]
    *[[Idempotent commutative groupoids]]
  *[[Idempotent groupoids]]

$\langle 2,0\rangle$ [[Pointed groupoids]]
  *[[Monoids]]
    *[[Commutative monoids]]
      *[[Bounded semilattices]]
    *[[Idempotent monoids]]
  *[[BCI-algebras]]*
    *[[BCK-algebras]]*
      *[[Commutative BCK-algebras]]

$\langle 2,1\rangle$ [[Groupoids with a unary operation]]
  *[[Inverse semigroups]]
    *[[Commutative inverse semigroups]]

$\langle 2,1,0\rangle$ [[Pointed groupoids with a unary operation]]
  *[[Groups]]
    *[[Abelian groups]]
      *[[Boolean groups]]

$\langle 2,1,0,1,1,\ldots\rangle$ [[Pointed groupoids with a unary operations]]
  *[[Modules over a ring]]
    *[[Vector spaces over a field]]

$\langle 2,2\rangle$ [[Duo-groupoids]]
  *[[Weakly associative lattices]]
    *[[Lattices]]
      *[[Modular lattices]]
        *[[Distributive lattices]]
      *[[Neardistributive lattices]]
        *[[Almost distributive lattices]]
  *[[Semirings]]
    *[[Idempotent semirings]]
  *[[Skew_lattices]]

$\langle 2,2,0\rangle$ [[Pointed duo-groupoids]]
  *[[Semirings with identity]]
  *[[Semirings with zero]]

$\langle 2,2,1\rangle$
  *[[Lattices with a unary operation]]
    *[[Distributive lattices with a unary operation]]
      *[[Distributive lattices with a unary operator]]
    *[[Lattices with a unary operator]]

$\langle 2,2,\ldots\rangle$
  *[[Lattices with additional operations]]
    *[[Distributive lattices with additional operations]]
      *[[Distributive lattices with operators]]
    *[[Lattices with operators]]

$\langle 2,0,2,0\rangle$
  *[[Shells]]
    *[[Bounded lattices]]
      *[[Bounded modular lattices]]
        *[[Bounded distributive lattices]]
  *[[Semirings with identity and zero]]
    *[[Boolean rings]]

$\langle 2,1,0,2\rangle$
  *[[Near-rings]]
    *[[Rings]]
      *[[Commutative rings]]

$\langle 2,1,0,2,0\rangle$
  *[[Rings with identity]]
    *[[Commutative rings with identity]]

$\langle 2,0,2,0,1\rangle$
  *[[Kleene algebras]]*
  *[[Bounded lattices with a unary operation]]
    *[[Bounded distributive lattices with a unary operation]]
      *[[Bounded distributive lattices with a unary operator]]
    *[[Bounded lattices with a unary operator]]
    *[[p-algebras]]
      *[[distributive p-algebras]]
        *[[Stone algebras]]
          *[[Double Stone algebras]]
            *[[Boolean algebras]]
    *[[dual p-algebras]]
      *[[distributive dual p-algebras]]

$\langle 2,0,2,0,1,1\rangle$
  *[[Boolean algebras with a unary operation]]
    *[[Modal algebras]]
      *[[Closure algebras]]
        *[[Monadic algebras]]

$\langle 2,0,2,0,1,1\rangle$
  *[[Boolean algebras with two unary operations]]
    *[[Boolean algebras with two unary operators]]
      *[[Tense algebras]]

$\langle 2,0,2,0,1,2\rangle$
  *[[Boolean algebras with a binary operation]]
    *[[Boolean algebras with a binary operator]]
      *[[Boolean semigroups]]
        *[[Commutative Boolean semigroups]]
          *[[Boolean semilattices]]

$\langle 2,0,2,0,1,2,0\rangle$
  *[[Boolean monoids]]
    *[[Commutative Boolean monoids]]
      *[[Symmetric relation algebras]]

$\langle 2,0,2,0,1,2,1,0\rangle$
  *[[Nonassociative relation algebras]]
    *[[Weakly associative relation algebras]]
      *[[Semiassociative relation algebras]]
        *[[Relation algebras]]
          *[[Representable relation algebras]]
            *[[Group relation algebras]]
              *[[Abelian group relation algebras]]

$\langle 2,0,2,0,1,2,0,2,2\rangle$
  *[[Sequential algebras]]
    *[[Representable sequential algebras]]

$\langle 2,0,2,0,1,\ldots\rangle$
  *[[Boolean algebras with additional operations]]
    *[[Boolean algebras with operators]]
      *[[Boolean modules over a relation algebra]]
      *[[Cylindric algebras of dimension n]]
        *[[Representable cylindric algebras of dimension n]]

$\langle 2,0,2,0,\ldots\rangle$
  *[[Heyting algebras]]

$\langle 2,0,2,0,\ldots\rangle$
  *[[Bounded lattices with additional operations]]
    *[[Bounded distributive lattices with additional operations]]
      *[[Bounded distributive lattices with operators]]
    *[[Bounded lattices with operators]]

$\langle 2,2,2\rangle$
  *[[Quasigroups]]

$\langle 2,2,2,0\rangle$
  *[[Loops]]
  *[[BCK-lattices]]
  *[[Brouwerian lattices]]

$\langle 2,2,2,1,0\rangle$
  *[[Lattice-ordered groups]]
    *[[Abelian lattice-ordered groups]]

$\langle 2,2,2,0,2\rangle$
  *[[Commutative residuated lattices]]
    *[[Commutative distributive residuated lattices]]
      *[[Commutative generalized BL-algebras]]
        *[[Commutative generalized MV-algebras]]

$\langle 2,2,2,0,2,2\rangle$
  *[[Residuated lattices]]
    *[[Cancellative residuated lattices]]
    *[[Distributive residuated lattices]]
      *[[Generalized BL-algebras]]
        *[[Generalized MV-algebras]]

$\langle 2,0,2,0,2,2\rangle$
  *[[Basic logic algebras]]

$\langle 2,0,2,0,2,0,2\rangle$
  *[[FLe-algebras]]
    *[[FLec-algebras]]
    *[[FLew-algebras]]

$\langle 2,0,2,0,2,0,2,2\rangle$
  *[[FL-algebras]]
    *[[FLc-algebras]]
    *[[FLw-algebras]]

$\langle 2,0,2,0,1,2,2\rangle$
  *[[Action algebras]]

$\langle 2,2,0,2,0,1,2,2\rangle$
  *[[Action lattices]]