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

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