General algebraic systems Varieties Concept
Variety of algebras
Summary
A variety of algebras is the class of all algebras of a fixed signature that satisfy a given set of identities. Groups, rings, lattices and modules over a fixed ring are the standard examples. Varieties are the basic objects of universal algebra, because many constructions, such as free algebras and presentations, work in every variety at once. By Birkhoff's theorem, a class of algebras is a variety exactly when it is closed under homomorphic images, subalgebras and direct products.
Where this leads
48 pages build on this one.
- Finitely based algebra
- Park's conjecture
- Evans's theorem on word problems
- Arithmetical variety
- Free algebra in a variety
- Baker's finite basis theorem
- Universal property of free algebras
- Interpretation of varieties
- Siggers term
- Birkhoff's HSP theorem
- Free spectrum of a variety
- Free product of algebras
- Variety as a reflective subcategory
- Near-unanimity term
- Mal'cev condition
- Word problem for an algebra
- Presentation of an algebra
- Tarski's finite basis problem
- Word problem for free algebras of a variety
- Discriminator variety
- Majority term
- Pixley term
- Minimal variety
- Equivalence of varieties
- Varieties as categories of algebras over a monad
- Amalgamated free product
- Post algebra
- Congruence-distributive variety
- Taylor term
- Jónsson terms
- Subvariety
- Inherently nonfinitely based algebra
- Relatively free algebra
- Equational basis
- Free functor
- McKenzie's undecidability of the finite basis problem
- Tarski's HSP formula
- Congruence-modular variety
- Finitely generated variety
- Foster's theorem on primal algebras
- Congruence-permutable variety
- Stone-type duality for varieties
- Residually small variety
- Day terms
- Locally finite variety
- Mal'cev term
- Lattice of subvarieties
- Gumm terms
Further reading
About this page
| Tag | AJNZJD8442 |
|---|---|
| URI | m2://AJNZJD8442 |
| Kind | concept |
| Writing | W0 stub |
| MSC | 08B99, 03C05 |
| Same as | |
| Changed | 2 November 2026 |
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Used by
- Finitely based algebra
- Park's conjecture
- Evans's theorem on word problems
- Arithmetical variety
- Free algebra in a variety
- Baker's finite basis theorem
- Universal property of free algebras
- Interpretation of varieties
- Siggers term
- Birkhoff's HSP theorem
- Free spectrum of a variety
- Free product of algebras
- Variety as a reflective subcategory
- Near-unanimity term
- Mal'cev condition
- Word problem for an algebra
- Presentation of an algebra
- Word problem for free algebras of a variety
- Discriminator variety
- Majority term
- Pixley term
- Minimal variety
- Equivalence of varieties
- Varieties as categories of algebras over a monad
- Amalgamated free product
- Post algebra
- Congruence-distributive variety
- Taylor term
- Jónsson terms
- Subvariety
- Inherently nonfinitely based algebra
- Relatively free algebra
- Equational basis
- Free functor
- Tarski's HSP formula
- Congruence-modular variety
- Finitely generated variety
- Foster's theorem on primal algebras
- Congruence-permutable variety
- Stone-type duality for varieties
- Residually small variety
- Day terms
- Locally finite variety
- Mal'cev term
- Lattice of subvarieties
- Gumm terms
Special cases
- Congruence-permutable variety
- Locally finite variety
- Congruence-distributive variety
- Minimal variety
- Finitely generated variety
- Congruence-modular variety
Results about this
- Birkhoff's HSP theorem
- Variety as a reflective subcategory
- Varieties as categories of algebras over a monad
- Tarski's HSP formula