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| + | ===== Mathematical Structures ===== | ||
| + | |||
| + | ====These pages have moved to GitHub==== | ||
| + | |||
| + | https:// | ||
| + | |||
| + | The pages 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 have moved. Click the link above for the latest version. /*are currently still under construction. Knowledgeable readers are encouraged to email additions or corrections to Peter Jipsen add or correct information.*/ | ||
| + | /*To enable the edit button on each page, use the Login link (above) to log in or create an account.*/ | ||
| + | |||
| + | 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 and would like some information added, please email Peter Jipsen (jipsen@chapman.edu). [Account creation unfortunately had to be disabled.] /*feel | ||
| + | free to add some relevant information and references by using the edit | ||
| + | link on the respective page. Pages are written in DokuWiki syntax, but mathematical notation is in standard LaTeX. | ||
| + | The previous | ||
| + | versions of these pages are still available at http:// | ||
| + | You can use this [[Sandbox]] page to try out editing (without worrying about deleting | ||
| + | useful information).*/ | ||
| + | |||
| + | / | ||
| + | |||
| + | **New:** Boldface links below display lists of finite algebras in a particular class (calculated locally in the browser). | ||
| + | |||
| + | /*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]] | ||
| + | |||
| + | [[Algebras]] | [[Logics]] | [[Syntax]] | [[Terms]] | [[Equations]] | [[Horn formulas]] | [[Universal formulas]] | [[First-order formulas]] /*| [[Theories]]*/ | ||
| + | |||
| + | === Alphabetical list of all classes === | ||
| + | -[[Abelian groups]] nbsp [[wp> | ||
| + | -[[Abelian lattice-ordered groups]] | ||
| + | -[[Abelian ordered groups]] | ||
| + | -[[Abelian p-groups]] | ||
| + | -[[Abelian partially ordered groups]] | ||
| + | -[[Action algebras]] nbsp [[wp> | ||
| + | -[[Action lattices]] | ||
| + | -[[Algebraic lattices]] | ||
| + | -[[Algebraic posets]] nbsp [[wp> | ||
| + | -[[Algebraic semilattices]] | ||
| + | -[[Allegories]] nbsp [[wp> | ||
| + | -[[Almost distributive lattices]] | ||
| + | -[[Associative algebras]] nbsp [[wp> | ||
| + | -[[Banach spaces]] nbsp [[wp> | ||
| + | -[[Bands]] nbsp [[wp> | ||
| + | -[[Basic logic algebras]] | ||
| + | -[[BCI-algebras]] nbsp [[wp>BCI algebra]] | ||
| + | -[[BCK-algebras]] nbsp [[wp>BCK algebra]] | ||
| + | -[[BCK-join-semilattices]] | ||
| + | -[[BCK-lattices]] | ||
| + | -[[BCK-meet-semilattices]] | ||
| + | -[[Bilattices]] | ||
| + | -[[Bilinear algebras]] | ||
| + | -[[Binars]], | ||
| + | -[[Basic logic algebras|BL-algebras]] | ||
| + | -[[Boolean algebras]] nbsp [[wp> | ||
| + | -[[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 partial monoids]] | ||
| + | -[[Cancellative semigroups]] | ||
| + | -[[Cancellative residuated lattices]] | ||
| + | -[[Categories]] | ||
| + | -[[Chains]] | ||
| + | -[[Clifford semigroups]] | ||
| + | -[[wp> | ||
| + | -[[Closure algebras]] | ||
| + | -[[Commutative BCK-algebras]] | ||
| + | -[[Commutative binars]], **[[Finite commutative binars]], [[Finite commutative binars with identity|with identity]], [[Finite commutative binars with zero|with zero]], [[Finite commutative binars with identity and zero|with identity and zero]]** | ||
| + | -[[Commutative idempotent involutive residuated lattices]] | ||
| + | -[[Commutative integral ordered monoids]], **[[Finite commutative integral ordered monoids]]** | ||
| + | -[[Commutative inverse semigroups]] | ||
| + | -[[Commutative involutive FL-algebras]] | ||
| + | -[[Commutative involutive residuated posets]] | ||
| + | -[[Commutative lattice-ordered monoids]] | ||
| + | -[[Commutative lattice-ordered rings]] | ||
| + | -[[Commutative lattice-ordered semigroups]] | ||
| + | -[[Commutative monoids]], **[[Finite commutative monoids]], [[Finite commutative monoids with zero]]** | ||
| + | -[[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]] | ||
| + | -[[wp> | ||
| + | -[[Conjugative binars]] | ||
| + | -[[Continuous lattices]] | ||
| + | -[[Continuous posets]] | ||
| + | -[[Cyclic involutive FL-algebras]] | ||
| + | -[[Cyclic involutive residuated 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 involutive residuated lattices]] | ||
| + | -[[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]] | ||
| + | -[[Effect algebras]] | ||
| + | -[[Medial groupoids|Entropic groupoids]] | ||
| + | -[[Equivalence algebras]] | ||
| + | -[[Equivalence relations]] | ||
| + | -[[Euclidean domains]] | ||
| + | -[[Function rings|f-rings]] | ||
| + | -[[Fields]] | ||
| + | -[[FL-algebras]] | ||
| + | -[[FLc-algebras]] | ||
| + | -[[Heyting algebras|FLcw-algebras]] | ||
| + | -[[FLe-algebras]] | ||
| + | -[[FLec-algebras]] | ||
| + | -[[Heyting Algebras|FLecw-algebras]] | ||
| + | -[[FLew-algebras]] | ||
| + | -[[FLi-algebras]] | ||
| + | -[[FLo-algebras]] | ||
| + | -[[FLw-algebras]] | ||
| + | -[[Frames]] | ||
| + | -[[Function rings]] | ||
| + | -[[G-sets]] | ||
| + | -[[Generalized BL-algebras]] | ||
| + | -[[Generalized Boolean algebras]] | ||
| + | -[[Generalized effect algebras]] | ||
| + | -[[Generalized MV-algebras]] | ||
| + | -[[Generalized orthoalgebras]] | ||
| + | -[[Generalized pseudo-effect algebras]] | ||
| + | -[[Generalized separation algebras]] | ||
| + | -[[Gödel algebras]] | ||
| + | -[[Graphs]] | ||
| + | -[[Groupoids]] | ||
| + | -[[Groups]] | ||
| + | -[[Hausdorff spaces]] | ||
| + | -[[Heyting algebras]] | ||
| + | -[[Hilbert algebras]] | ||
| + | -[[Hilbert spaces]] | ||
| + | -[[Hoops]] | ||
| + | -[[Idempotent residuated lattices]] | ||
| + | -[[Idempotent involutive residuated lattices]] | ||
| + | -[[Idempotent semirings]] | ||
| + | -[[Idempotent semirings with identity]] | ||
| + | -[[Idempotent semirings with identity and zero]] | ||
| + | -[[Idempotent semirings with zero]] | ||
| + | -[[Tarski algebras|Implication algebras]] | ||
| + | -[[Implicative lattices]] | ||
| + | -[[Integral domains]] | ||
| + | -[[Integral involutive FL-algebras]] | ||
| + | -[[Integral ordered monoids]], **[[finite integral ordered monoids]]** | ||
| + | -[[Integral relation algebras]] | ||
| + | -[[Integral residuated lattices]] | ||
| + | -[[Intuitionistic linear logic algebras]] | ||
| + | -[[Inverse semigroups]] | ||
| + | -[[Involutive FL-algebras]] | ||
| + | -[[Involutive lattices]] | ||
| + | -[[Involutive FL-algebras|Involutive residuated lattices]] | ||
| + | -[[Involutive residuated posets]] | ||
| + | -[[Join-semidistributive lattices]] | ||
| + | -[[Semilattices|Join-semilattices]] | ||
| + | -[[Jordan algebras]] | ||
| + | -[[Kleene algebras]] | ||
| + | -[[Kleene lattices]] | ||
| + | -[[Kleene logic algebras]] | ||
| + | -[[Residuated partially ordered semigroups|Lambek algebras]] | ||
| + | -[[Lattice effect algebras]] | ||
| + | -[[Lattice implication algebras]] | ||
| + | -[[Lattice-ordered groups]] | ||
| + | -[[Lattice-ordered monoids]] | ||
| + | -[[Lattice-ordered rings]] | ||
| + | -[[Lattice-ordered semigroups]] | ||
| + | -[[Lattices]] | ||
| + | -[[Left cancellative semigroups]] | ||
| + | -[[Left neofield]] | ||
| + | -[[Lie algebras]] | ||
| + | -[[Commutative residuated partially ordered monoids|Lineales]] | ||
| + | -[[Goedel algebras|Linear Heyting algebras]] | ||
| + | -[[Linear logic algebras]] | ||
| + | -[[Chains|Linear orders]] | ||
| + | -[[Frames|Locales]] | ||
| + | -[[wp> | ||
| + | -[[Loops]] | ||
| + | -[[Lukasiewicz algebras of order n]] | ||
| + | -[[M-sets]] | ||
| + | -[[Binars|Magmas]] | ||
| + | -[[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]], | ||
| + | -[[Monounary algebras]] | ||
| + | -[[Moufang loops]] | ||
| + | -[[Moufang quasigroups]] | ||
| + | -[[Multiplicative additive linear logic algebras]] | ||
| + | -[[Multiplicative lattices]] | ||
| + | -[[Multiplicative semilattices]] | ||
| + | -[[Multisets]] | ||
| + | -[[MV-algebras]] | ||
| + | -[[MV-effect algebras]] | ||
| + | -[[m-zeroids]] | ||
| + | -[[Neardistributive lattices]] | ||
| + | -[[Near-rings]] | ||
| + | -[[Near-rings with identity]] | ||
| + | -[[Near-fields]] | ||
| + | -[[Neofields]] | ||
| + | -[[Nilpotent groups]] | ||
| + | -[[Nonassociative relation algebras]] | ||
| + | -[[Bilinear algebras|Nonassociative algebras]] | ||
| + | -[[Normal bands]] | ||
| + | -[[Normal valued lattice-ordered groups]] | ||
| + | -[[Normed vector spaces]] | ||
| + | -[[Ockham algebras]] | ||
| + | -[[Order algebras]] | ||
| + | -[[Ordered abelian groups]] | ||
| + | -[[Ordered fields]] | ||
| + | -[[Ordered groups]] | ||
| + | -[[Ordered monoids]] | ||
| + | -[[Ordered monoids with zero]] | ||
| + | -[[Ordered rings]] | ||
| + | -[[Ordered semigroups]], | ||
| + | -[[Ordered semilattices]], | ||
| + | -[[Partially ordered sets|Ordered sets]] | ||
| + | -[[Ore domains]] | ||
| + | -[[Orthoalgebras]] | ||
| + | -[[Ortholattices]] | ||
| + | -[[Orthomodular lattices]] | ||
| + | -[[p-groups]] | ||
| + | -[[Partial groupoids]] | ||
| + | -[[Partial monoid]] | ||
| + | -[[Partial semigroups]] | ||
| + | -[[Partially ordered groups]] | ||
| + | -[[Partially ordered monoids]] | ||
| + | -[[Partially ordered semigroups]] | ||
| + | -[[Partially ordered sets]] | ||
| + | -[[Peirce algebras]] | ||
| + | -[[Pocrims]] | ||
| + | -[[FL-algebras|Pointed residuated lattices]] | ||
| + | -[[Polrims]] | ||
| + | -[[wp> | ||
| + | -[[Partially ordered sets|Posets]] | ||
| + | -[[Post algebras]] | ||
| + | -[[Preordered sets]] | ||
| + | -[[Priestley spaces]] | ||
| + | -[[Principal Ideal Domains]] | ||
| + | -[[Process algebras]] | ||
| + | -[[Pseudo basic logic algebras]] | ||
| + | -[[Pseudo-effect algebras]] | ||
| + | -[[Pseudo MTL-algebras]] | ||
| + | -[[Pseudo MV-algebras]] | ||
| + | -[[Pseudocomplemented distributive lattices]] | ||
| + | -[[Pure discriminator algebras]] | ||
| + | -[[Quandles]] | ||
| + | -[[Quantales]] | ||
| + | -[[Quasigroups]] | ||
| + | -[[Quasi-implication algebras]] | ||
| + | -[[Quasi-MV-algebra]] | ||
| + | -[[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]] | ||
| + | -[[Residuated partially ordered monoids|Residuated posets]] | ||
| + | -[[Right residuated binars]] | ||
| + | -[[Right hoops]] | ||
| + | -[[Right quasigroups]] | ||
| + | -[[Rings]] | ||
| + | -[[Rings with identity]] | ||
| + | -[[Schroeder categories]] | ||
| + | -[[Semiassociative relation algebras]] | ||
| + | -[[Semidistributive lattices]] | ||
| + | -[[Semifields]] | ||
| + | -[[Semigroups]], | ||
| + | -[[Monoids|Semigroups with identity]] | ||
| + | -[[Semigroups with zero]], **[[Finite semigroups with zero]]** | ||
| + | -[[Semilattices]], | ||
| + | -[[Semilattices with identity]], **[[Finite semilattices with identity]]** | ||
| + | -[[Semilattices with zero]] | ||
| + | -[[Semirings]] | ||
| + | -[[Semirings with identity]] | ||
| + | -[[Semirings with identity and zero]] | ||
| + | -[[Semirings with zero]] | ||
| + | -[[Separation algebras]] | ||
| + | -[[Sequential algebras]] | ||
| + | -[[Sets]] | ||
| + | -[[Shells]] | ||
| + | -[[Division rings|Skew-fields]] | ||
| + | -[[Skew lattices]] | ||
| + | -[[Small categories]] | ||
| + | -[[Sober T0-spaces]] | ||
| + | -[[Solvable groups]] | ||
| + | -[[Sqrt-quasi-MV-algebras]] | ||
| + | -[[Stably compact spaces]] | ||
| + | -[[Steiner quasigroups]] | ||
| + | -[[Stone algebras]] | ||
| + | -[[Sugihara algebras]] | ||
| + | -[[Sugihara monoids]] | ||
| + | -[[Symmetric relations]] | ||
| + | -[[T0-spaces]] | ||
| + | -[[T1-spaces]] | ||
| + | -[[Hausdorff spaces|T2-spaces]] | ||
| + | -[[Tarski algebras]] | ||
| + | -[[Tense algebras]] | ||
| + | -[[Temporal algebras]] | ||
| + | -[[Topological groups]] | ||
| + | -[[Topological spaces]] | ||
| + | -[[wp> | ||
| + | -[[Torsion groups]] | ||
| + | -[[Abelian ordered groups|Totally ordered abelian groups]] | ||
| + | -[[Ordered groups|Totally ordered 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]] | ||
