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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.// | ||

+ | |||

+ | Initially the main content concerns mostly first-order classes of relational | ||

+ | structures and, more particularly, | ||

+ | 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 http:// | ||

+ | 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]] | ||

+ | -[[Basic logic algebras|BL-algebras]] | ||

+ | -[[Boolean algebras]] | ||

+ | -[[Boolean algebras with operators]] | ||

+ | -[[Boolean groups]] | ||

+ | -[[Boolean lattices]] | ||

+ | -[[Boolean modules over a relation algebra]] | ||

+ | -[[Boolean monoids]] | ||

+ | -[[Boolean algebras|Boolean rings]] | ||

+ | -[[Boolean semigroups]] | ||

+ | -[[Boolean semilattices]] | ||

+ | -[[Boolean spaces]] | ||

+ | -[[Bounded distributive lattices]] | ||

+ | -[[Bounded lattices]] | ||

+ | -[[Bounded residuated lattices]] | ||

+ | -[[Brouwerian algebras]] | ||

+ | -[[Brouwerian semilattices]] | ||

+ | -[[wp> | ||

+ | -[[Cancellative commutative monoids]] | ||

+ | -[[Cancellative commutative semigroups]] | ||

+ | -[[Cancellative monoids]] | ||

+ | -[[Cancellative semigroups]] | ||

+ | -[[Cancellative residuated lattices]] | ||

+ | -[[Categories]] | ||

+ | -[[Chains]] | ||

+ | -[[Clifford semigroups]] | ||

+ | -[[wp> | ||

+ | -[[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]] | ||

+ | -[[Boolean spaces|Compact zero-dimensional Hausdorff spaces]] | ||

+ | -[[Complemented lattices]] | ||

+ | -[[Boolean lattices|Complemented distributive lattices]] | ||

+ | -[[Complemented modular lattices]] | ||

+ | -[[Complete distributive lattices]] | ||

+ | -[[Complete lattices]] | ||

+ | -[[Complete semilattices]] | ||

+ | -[[Directed complete partial orders|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]] | ||

+ | -[[Medial groupoids|Entropic groupoids]] | ||

+ | -[[Equivalence algebras]] | ||

+ | -[[Equivalence relations]] | ||

+ | -[[Euclidean domains]] | ||

+ | -[[Function rings|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]] | ||

+ | -[[Semilattices|Join-semilattices]] | ||

+ | -[[Jordan algebras]] | ||

+ | -[[Kleene algebras]] | ||

+ | -[[Kleene lattices]] | ||

+ | -[[Residuated partially ordered semigroups|Lambek algebras]] | ||

+ | -[[Lattice-ordered groups]] | ||

+ | -[[Lattice-ordered monoids]] | ||

+ | -[[Lattice-ordered rings]] | ||

+ | -[[Lattice-ordered semigroups]] | ||

+ | -[[Lattices]] | ||

+ | -[[Left cancellative semigroups]] | ||

+ | -[[Lie algebras]] | ||

+ | -[[Goedel algebras|Linear Heyting algebras]] | ||

+ | -[[Linear logic algebras]] | ||

+ | -[[Chains|Linear orders]] | ||

+ | -[[Frames|Locales]] | ||

+ | -[[wp> | ||

+ | -[[Loops]] | ||

+ | -[[Lukasiewicz algebras of order n]] | ||

+ | -[[M-sets]] | ||

+ | -[[Medial groupoids]] | ||

+ | -[[Medial quasigroups]] | ||

+ | -[[Meet-semidistributive lattices]] | ||

+ | -[[Semilattices|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]] | ||

+ | -[[Bilinear 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]] | ||

+ | -[[Partially ordered sets|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]] | ||

+ | -[[FL-algebras|Pointed residuated lattices]] | ||

+ | -[[Polrims]] | ||

+ | -[[Polyadic algebras]] | ||

+ | -[[Partially ordered sets|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]] | ||

+ | -[[Preordered sets|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]] | ||

+ | -[[Monoids|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]] | ||

+ | -[[Division rings|Skew-fields]] | ||

+ | -[[Skew_lattices]] | ||

+ | -[[Small categories]] | ||

+ | -[[Sober T0-spaces]] | ||

+ | -[[Solvable groups]] | ||

+ | -[[Stably compact spaces]] | ||

+ | -[[Steiner quasigroups]] | ||

+ | -[[Stone algebras]] | ||

+ | -[[Symmetric relations]] | ||

+ | -[[T0-spaces]] | ||

+ | -[[T1-spaces]] | ||

+ | -[[Hausdorff spaces|T2-spaces]] | ||

+ | -[[Tarski algebras]] | ||

+ | -[[Tense algebras]] | ||

+ | -[[Temporal algebras]] | ||

+ | -[[Topological groups]] | ||

+ | -[[Topological spaces]] | ||

+ | -[[wp> | ||

+ | -[[Torsion groups]] | ||

+ | -[[Ordered monoids|Totally ordered monoids]] | ||

+ | -[[Transitive relations]] | ||

+ | -[[Trees]] | ||

+ | -[[Tournaments]] | ||

+ | -[[Unary algebras]] | ||

+ | -[[Unique factorization domains]] | ||

+ | -[[Rings with identity|Unital rings]] | ||

+ | -[[Vector spaces]] | ||

+ | -[[MV-algebras|Wajsberg algebras]] | ||

+ | -[[Wajsberg hoops]] | ||

+ | -[[Weakly associative lattices]] | ||

+ | -[[Weakly associative relation algebras]] | ||

+ | -[[Weakly representable relation algebras]] | ||

+ | |||

+ | [[Template]] | ||