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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 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, equationally defined classes of algebraic structures.
 +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://math.chapman.edu/cgi-bin/structures.pl ).
 +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>C-star-algebras]]
 +  -[[Cancellative commutative monoids]]
 +  -[[Cancellative commutative semigroups]]
 +  -[[Cancellative monoids]]
 +  -[[Cancellative semigroups]]
 +  -[[Cancellative residuated lattices]]
 +  -[[Categories]]
 +  -[[Chains]]
 +  -[[Clifford semigroups]]
 +  -[[wp>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]]
 +  -[[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>Locally compact topological spaces]]
 +  -[[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>Topological vector spaces]]
 +  -[[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]]
  

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