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The webpages collected here list information about classes of
mathematical structures. The aim is to have a central place to check
what properties are known about these structures.
Initially the main content concerns first-order classes of relational
structures and, more particularly, varieties of universal algebras.
If you are familiar with some of these classes of algebras, feel
free to add some relevant information and references by using the edit
link at the bottom of the respective page. A phrase followed
by a blue '?' is a link to a page that still needs to be filled in. Adding and
formatting pages is really simple. Here are some basic [text
formatting rules] (see also Wikipedia[how to edit a page],
although not all the features mentioned there are available here).
Mathematical notation is delimited by dollar signs, and uses a very basic subset of LaTeX
(essentially those characters and symbols that can be displayed by most browsers).
You can use this sandbox page to try out editing (without worrying about deleting
New: There is now an experimental XHTML+MathML version of these pages at http://math.chapman.edu/cgi-bin/structuresxml.pl (go to [Display MathML on these pages] to check if your browser can handle MathML formulas).
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Please use this template
for new pages and follow the format of
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the value of a property should be justified by a brief reason or a
reference to a publication where the result can be found. Bibtex references
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using the [bibtex2xml converter].
This front page can also be edited to add further classes of structures. A criterion for inclusion
in this list is that there should be some journal publication or book where the class has been
named and defined.
Acknowledgements | Notation and terminology | Properties | Tools | Online books and lecture notes
Varieties | Quasivarieties | Universal classes | First-order classes | Second-order classes
Classes of ordered algebraic structures | Partial algebras | Many-sorted algebras | Logics
Alphabetical list of all classes
- Abelian groups
- Abelian lattice-ordered groups
- Action algebras
- Action lattices
- Algebraic lattices
- Algebraic posets
- Algebraic semilattices
- Almost distributive lattices
- [Associative algebras]
- [Banach spaces]
- Basic logic algebras
- Boolean algebras
- Boolean algebras with operators
- Boolean groups
- Boolean lattices
- Boolean modules over a relation algebra
- Boolean monoids
- Boolean rings
- Boolean semigroups
- Boolean semilattices
- Boolean spaces
- Bounded distributive lattices
- Bounded lattices
- [Bounded residuated lattices]?
- Brouwerian algebras
- Brouwerian semilattices
- Cancellative commutative monoids
- Cancellative commutative semigroups
- Cancellative monoids
- Cancellative semigroups
- Cancellative residuated lattices
- Clifford semigroups
- [Clifford algebras]
- Closure algebras
- Commutative BCK-algebras
- Commutative groupoids
- Commutative inverse semigroups
- Commutative monoids
- Commutative regular rings
- Commutative residuated lattices
- Commutative rings
- Commutative rings with identity
- Commutative semigroups
- [Compact topological spaces]
- [Compact zero-dimensional Hausdorff spaces]?
- Complemented lattices
- Complemented distributive lattices
- Complemented modular lattices
- [Complete distributive lattices]?
- Complete lattices
- Complete semilattices
- Complete partial orders
- [Completely regular Hausdorff spaces]?
- Completely regular semigroups
- [Continuous lattices]?
- [Continuous posets]?
- [Cylindric algebras]?
- De Morgan algebras
- [De Morgan monoids]?
- [Dedekind categories]?
- [Dedekind domains]?
- Dense linear orders
- [Digraph algebras]?
- Directed complete partial orders
- Directed partial orders
- [Directed graphs]?
- [Distributive allegories]?
- Distributive double p-algebras
- Distributive dual p-algebras
- Distributive lattices
- [Distributive lattices with operations]?
- [Distributive lattice ordered semigroups]?
- Distributive p-algebras
- Distributive residuated lattices
- [Division algebras]?
- Division rings
- Double Stone algebras
- [Dunn monoids]?
- [Dynamic algebras]?
- Entropic groupoids
- [Equivalence algebras]?
- Euclidean domains
- Function rings
- Generalized BL-algebras
- Generalized Boolean algebras
- Generalized MV-algebras
- Goedel algebras
- [Halmos algebras]?
- Hausdorff spaces
- Heyting algebras
- [Hilbert algebras]?
- Hilbert spaces
- Idempotent semirings
- Idempotent semirings with identity
- Idempotent semirings with identity and zero
- Idempotent semirings with zero
- Implicative lattices
- Integral domains
- [Integral relation algebras]?
- [Integral residuated lattices]?
- [Intuitionistic linear logic algebras]?
- Inverse semigroups
- Involutive lattices
- [Involutive residuated lattices]?
- Join-semidistributive lattices
- [Jordan algebras]?
- Kleene algebras
- Kleene lattices
- [Lambek algebras]?
- Lattice-ordered groups
- Lattice-ordered rings
- Left cancellative semigroups
- [Lie algebras]?
- Linear Heyting algebras
- [Linear logic algebras]?
- Linear orders
- [Locally compact topological spaces]
- Lukasiewicz algebras of order n
- Medial groupoids
- [Medial quasigroups]?
- Meet-semidistributive lattices
- Metric spaces
- Modal algebras
- Modular lattices
- [Modular ortholattices]?
- Modules over a ring
- [Monadic algebras]?
- [Monoidal t-norm logic algebras]?
- Moufang loops
- [Moufang quasigroups]?
- [Multiplicative additive linear logic algebras]?
- [Multiplicative lattices]?
- Neardistributive lattices
- Near-rings with identity
- [Nilpotent groups]?
- Nonassociative relation algebras
- Normal bands
- [Normed vector spaces]?
- Ockham algebras
- Order algebras
- [Ordered fields]?
- [Ordered rings]?
- Ordered sets
- [Ore domains]?
- Orthomodular lattices
- [Partial groupoids]?
- Partially ordered sets
- [Peirce algebras]?
- Pointed residuated lattices
- [Post algebras]?
- Preordered sets
- [Priestley spaces]?
- Principal Ideal Domains
- [Process algebras]?
- [Pseudo basic logic algebras]?
- [Pseudo MTL-algebras]?
- [Pseudo MV-algebras]?
- Pseudocomplemented distributive lattices
- [Pure discriminator algebras]?
- Quasi-ordered sets
- Rectangular bands
- Regular rings
- Regular semigroups
- Relation algebras
- [Relative Stone algebras]?
- [Relativized relation algebras]?
- [Representable cylindric algebras]?
- [Representable relation algebras]?
- Residuated lattices
- [Residuated idempotent semirings]?
- Rings with identity
- [Schroeder categories]?
- [Semiassociative relation algebras]?
- Semidistributive lattices
- Semigroups with identity
- Semigroups with zero
- Semilattices with identity
- Semilattices with zero
- Semirings with identity
- Semirings with identity and zero
- Semirings with zero
- Sequential algebras
- [Small categories]?
- [Sober T0-spaces]?
- [Solvable groups]?
- [Stably compact spaces]?
- [Steiner quasigroups]?
- Stone algebras
- [Tarski algebras]?
- Tense algebras
- [Temporal algebras]?
- [Topological groups]?
- Topological spaces
- [Topological vector spaces]
- [Torsion groups]?
- [Unary algebras]?
- Unique factorization domains
- Unital rings
- Vector spaces over a field
- Wajsberg algebras
- Wajsberg hoops
- [Weakly associative lattices]?
- [Weakly associative relation algebras]?
- [Weakly representable relation algebras]?