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Mathematical Structures

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.

'These pages are currently still under construction. Knowledgeable readers are encouraged to add or correct information.' To enable the edit link at the bottom of each page, use the Login link (above) to enter a Username of your choice and the password “mathstruct”.

Initially the main content concerns mostly first-order classes of relational structures and, more particularly, varieties of universal algebras. If you are familiar with some of these classes of structures, feel free to add some relevant information and references by using the edit link on the respective page. Most pages are written in standard LaTeX (if you don't know LaTeX you should probably not edit these pages – the previous non-LaTeX version of these pages is still available at ). You can use this Sandbox page to try out editing (without worrying about deleting useful information).

Suggestions or comments? Just follow the link and edit the page.

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 | Logics

Alphabetical list of all classes

-Abelian groups -Abelian lattice-ordered groups -Abelian ordered groups -Abelian p-groups -Abelian partially ordered groups -Action algebras -Action lattices -Algebraic lattices -Algebraic posets -Algebraic semilattices -Allegories -Almost distributive lattices -Associative algebras -Banach spaces -Bands -Basic logic algebras -BCI-algebras -BCK-algebras -BCK-join-semilattices -BCK-lattices -BCK-meet-semilattices -Bilinear algebras -BL-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 -C-star-algebras -Cancellative commutative monoids -Cancellative commutative semigroups -Cancellative monoids -Cancellative semigroups -Cancellative residuated lattices -Categories -Chains -Clifford semigroups -[ Clifford algebras] -Closure algebras -Commutative BCK-algebras -Commutative groupoids -Commutative inverse semigroups -Commutative lattice-ordered monoids -Commutative lattice-ordered rings -Commutative lattice-ordered semigroups -Commutative monoids -Commutative ordered monoids -Commutative ordered rings -Commutative ordered semigroups -Commutative partially ordered monoids -Commutative partially ordered semigroups -Commutative regular rings -Commutative residuated lattice-ordered semigroups -Commutative residuated lattices -Commutative residuated partially ordered monoids -Commutative residuated partially ordered semigroups -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 -Directoids -Distributive allegories -Distributive double p-algebras -Distributive dual p-algebras -Distributive lattice expansions -Distributive lattices -Distributive lattices with operators -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 -Equivalence relations -Euclidean domains -f-rings -Fields -FL-algebras -FLc-algebras -FLe-algebras -FLew-algebras -FLw-algebras -Frames -Function rings -G-sets -Generalized BL-algebras -Generalized Boolean algebras -Generalized MV-algebras -Goedel algebras -Graphs -Groupoids -Groups -Hausdorff spaces -Heyting algebras -Hilbert algebras -Hilbert spaces -Hoops -Idempotent semirings -Idempotent semirings with identity -Idempotent semirings with identity and zero -Idempotent semirings with zero -Implication algebras -Implicative lattices -Integral domains -Integral ordered monoids -Integral relation algebras -Integral residuated lattices -Intuitionistic linear logic algebras -Inverse semigroups -Involutive lattices -Involutive residuated lattices -Join-semidistributive lattices -Join-semilattices -Jordan algebras -Kleene algebras -Kleene lattices -Lambek algebras -Lattice-ordered groups -Lattice-ordered monoids -Lattice-ordered rings -Lattice-ordered semigroups -Lattices -Left cancellative semigroups -Lie algebras -Linear Heyting algebras -Linear logic algebras -Linear orders -Locales -[ Locally compact topological spaces] -Loops -Lukasiewicz algebras of order n -M-sets -Medial groupoids -Medial quasigroups -Meet-semidistributive lattices -Meet-semilattices -Metric spaces -Modal algebras -Modular lattices -Modular ortholattices -Modules over a ring -Monadic algebras -Monoidal t-norm logic algebras -Monoids -Moufang loops -Moufang quasigroups -Multiplicative additive linear logic algebras -Multiplicative lattices -Multiplicative semilattices -Multisets -MV-algebras -Neardistributive lattices -Near-rings -Near-rings with identity -Near-fields -Nilpotent groups -Nonassociative relation algebras -Nonassociative algebras -Normal bands -Normed vector spaces -Ockham algebras -Order algebras -Ordered fields -Ordered groups -Ordered monoids -Ordered monoids with zero -Ordered rings -Ordered semigroups -Ordered sets -Ore domains -Ortholattices -Orthomodular lattices -p-groups -Partial groupoids -Partial semigroups -Partially ordered groups -Partially ordered monoids -Partially ordered semigroups -Partially ordered sets -Peirce algebras -Pocrims -Pointed residuated lattices -Polrims -Polyadic algebras -Posets -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 -Quantales -Quasigroups -Quasi-implication algebras -Quasi-ordered sets -Quasitrivial groupoids -Rectangular bands -Reflexive relations -Regular rings -Regular semigroups -Relation algebras -Relative Stone algebras -Relativized relation algebras -Representable cylindric algebras -Representable lattice-ordered groups -Representable relation algebras -Representable residuated lattices -Residuated idempotent semirings -Residuated lattice-ordered semigroups -Residuated lattices -Residuated partially ordered monoids -Residuated partially ordered semigroups -Rings -Rings with identity -Schroeder categories -Semiassociative relation algebras -Semidistributive lattices -Semigroups -Semigroups with identity -Semigroups with zero -Semilattices -Semilattices with identity -Semilattices with zero -Semirings -Semirings with identity -Semirings with identity and zero -Semirings with zero -Sequential algebras -Sets -Shells -Skew-fields -Skew_lattices -Small categories -Sober T0-spaces -Solvable groups -Stably compact spaces -Steiner quasigroups -Stone algebras -Symmetric relations -T0-spaces -T1-spaces -T2-spaces -Tarski algebras -Tense algebras -Temporal algebras -Topological groups -Topological spaces -[ Topological vector spaces] -Torsion groups -Totally ordered monoids -Transitive relations -Trees -Tournaments -Unary algebras -Unique factorization domains -Unital rings -Vector spaces -Wajsberg algebras -Wajsberg hoops -Weakly associative lattices -Weakly associative relation algebras -Weakly representable relation algebras