Varieties

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 K of algebras is a variety iff it is closed under the operators H, S, P (i.e., HK ⊆ K, SK ⊆ K, and PK ⊆ K), where

HK = {homomorphic images of members of K}
SK = {subalgebras of members of K}
PK = {direct products of members of 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

() Sets

(0) [Pointed sets]?

(1) [Mono-unary algebras]?

(1,0) [Pointed mono-unary algebras]?

(1,1) [Duo-unary algebras]?

(1,1, . . .) [Unary algebras]?

• M-sets?
  • G-sets?

(2) Groupoids

• Semigroups
  • Commutative semigroups
    • Semilattices
  • Bands
    • Normal bands
      • Rectangular bands
• Commutative groupoids
  • [Idempotent commutative groupoids]?
• [Idempotent groupoids]?

(2,0) [Pointed groupoids]?

• Monoids
  • Commutative monoids
    • [Bounded semilattices]?
  • [Idempotent monoids]?
• BCI-algebras*
  • BCK-algebras*
    • Commutative BCK-algebras

(2,1) [Groupoids with a unary operation]?

• Inverse semigroups
  • Commutative inverse semigroups

(2,1,0) [Pointed groupoids with a unary operation]?

• Groups
  • Abelian groups
    • Boolean groups

(2,1,0,1,1, . . .) [Pointed groupoids with a unary operations]?

• Modules over a ring
  • Vector spaces over a field

(2,2) Duo-groupoids?

• [Weakly associative lattices]?
  • Lattices
    • Modular lattices
      • Distributive lattices
    • Neardistributive lattices
      • Almost distributive lattices
• Semirings
  • Idempotent semirings

(2,2,0) [Pointed duo-groupoids]?

• Semirings with identity
• Semirings with zero

(2,2,1)

• [Lattices with a unary operation]?
  • [Distributive lattices with a unary operation]?
    • [Distributive lattices with a unary operator]?
  • [Lattices with a unary operator]?

(2,2, . . .)

• [Lattices with additional operations]?
  • [Distributive lattices with additional operations]?
    • [Distributive lattices with operators]?
  • [Lattices with operators]?

(2,0,2,0)

• Shells
  • Bounded lattices
    • [Bounded modular lattices]?
      • Bounded distributive lattices
• Semirings with identity and zero
  • [Boolean rings]?

(2,1,0,2)

• Near-rings
  • Rings
    • Commutative rings

(2,1,0,2,0)

• Rings with identity
  • Commutative rings with identity

(2,0,2,0,1)

• 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

(2,0,2,0,1,1)

• [Boolean algebras with a unary operation]?
  • Modal algebras
    • Closure algebras
      • [Monadic algebras]?

(2,0,2,0,1,1)

• [Boolean algebras with two unary operations]?
  • [Boolean algebras with two unary operators]?
    • Tense algebras

(2,0,2,0,1,2)

• [Boolean algebras with a binary operation]?
  • [Boolean algebras with a binary operator]?
    • Boolean semigroups
      • [Commutative Boolean semigroups]?
        • Boolean semilattices

(2,0,2,0,1,2,0)

• Boolean monoids
  • [Commutative Boolean monoids]?
    • [Symmetric relation algebras]?

(2,0,2,0,1,2,1,0)

• Nonassociative relation algebras
  • [Weakly associative relation algebras]?
    • [Semiassociative relation algebras]?
      • Relation algebras
        • [Representable relation algebras]?
          • [Group relation algebras]?
            • [Abelian group relation algebras]?

(2,0,2,0,1,2,0,2,2)

• Sequential algebras
  • [Representable sequential algebras]?

(2,0,2,0,1, . . .)

• [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]?

(2,0,2,0, . . .)

• Heyting algebras

(2,0,2,0, . . .)

• [Bounded lattices with additional operations]?
  • [Bounded distributive lattices with additional operations]?
    • [Bounded distributive lattices with operators]?
  • [Bounded lattices with operators]?

(2,2,2)

• Quasigroups

(2,2,2,0)

• Loops
• BCK-lattices
• [Brouwerian lattices]?

(2,2,2,1,0)

• Lattice-ordered groups
  • Abelian lattice-ordered groups

(2,2,2,0,2)

• Commutative residuated lattices
  • Commutative distributive residuated lattices
    • [Commutative generalized BL-algebras]?
      • [Commutative generalized MV-algebras]?

(2,2,2,0,2,2)

• Residuated lattices
  • Cancellative residuated lattices
  • Distributive residuated lattices
    • Generalized BL-algebras
      • Generalized MV-algebras

(2,0,2,0,2,2)

• Basic logic algebras

(2,0,2,0,2,0,2)

• FLe-algebras
  • FLec-algebras?
  • FLew-algebras?

(2,0,2,0,2,0,2,2)

• FL-algebras
  • FLc-algebras?
  • FLw-algebras?

(2,0,2,0,1,2,2)

• Action algebras

(2,2,0,2,0,1,2,2)

• Action lattices