# Varieties

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### 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

Proper quasivarieties are marked by a *

() 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

(2,0) [Pointed groupoids]?

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

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

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

(2,2) Duo-groupoids?

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

(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, . . .)

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

(2,0,2,0)

(2,1,0,2)

(2,1,0,2,0)

(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?
• [dual p-algebras]?

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

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

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

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

• [Boolean algebras with a binary operation]?

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

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

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

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

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

(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)

(2,2,2,0)

(2,2,2,1,0)

(2,2,2,0,2)

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

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

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

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

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

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

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