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start [2021/02/22 21:10]
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 mathematical structures. The aim is to have a central place to check mathematical structures. The aim is to have a central place to check
 what properties are known about these structures.  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 Initially the main content concerns mostly first-order classes of relational
 structures and, more particularly, equationally defined classes of algebraic structures. structures and, more particularly, equationally defined classes of algebraic structures.
-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.+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).
-**New:** Boldface links below display lists of finite algebras in a particular class (calculated locally in the browser).+[[Suggestions or comments]]? Just follow the link and edit the page.
-[[Acknowledgements]] | [[Notation and terminology]] | [[Properties]] | [[Tools]] | [[Online books and lecture notes]]+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.
-[[Varieties]] | [[Quasivarieties]] | [[Universal classes]] | [[First-order classes]] | [[Second-order classes]] +[[Acknowledgements]] | [[Notation and terminology]] | [[Properties]] |  
 +[[Tools]] | [[Online books and lecture notes]]
-[[Algebras]] | [[Logics]] | [[Syntax]] | [[Terms]] | [[Equations]] | [[Horn formulas]] | [[Universal formulas]] | [[First-order formulas]] +[[Varieties]] | [[Quasivarieties]] | [[Universal classes]] | [[First-order classes]] |  
 +[[Second-order classes]] | [[Logics]]
 === Alphabetical list of all classes === === Alphabetical list of all classes ===
-  -[[Abelian groups]] [[wp>Abelian group]]+  -[[Abelian groups]]
   -[[Abelian lattice-ordered groups]]   -[[Abelian lattice-ordered groups]]
   -[[Abelian ordered groups]]   -[[Abelian ordered groups]]
   -[[Abelian p-groups]]   -[[Abelian p-groups]]
   -[[Abelian partially ordered groups]]   -[[Abelian partially ordered groups]]
-  -[[Action algebras]] [[wp>Action algebra]]+  -[[Action algebras]]
   -[[Action lattices]]   -[[Action lattices]]
   -[[Algebraic lattices]]   -[[Algebraic lattices]]
-  -[[Algebraic posets]] [[wp>Algebraic poset]]+  -[[Algebraic posets]]
   -[[Algebraic semilattices]]   -[[Algebraic semilattices]]
-  -[[Allegories]] [[wp>Allegory (category theory)]]+  -[[Allegories]]
   -[[Almost distributive lattices]]   -[[Almost distributive lattices]]
-  -[[Associative algebras]] [[wp>Associative algebra]] +  -[[Associative algebras]] 
-  -[[Banach spaces]] [[wp>Banach space]] +  -[[Banach spaces]] 
-  -[[Bands]] [[wp>Band (mathematics)]], **[[Finite bands]]**+  -[[Bands]]
   -[[Basic logic algebras]]   -[[Basic logic algebras]]
-  -[[BCI-algebras]] [[wp>BCI algebra]] +  -[[BCI-algebras]] 
-  -[[BCK-algebras]] [[wp>BCK algebra]]+  -[[BCK-algebras]]
   -[[BCK-join-semilattices]]   -[[BCK-join-semilattices]]
   -[[BCK-lattices]]   -[[BCK-lattices]]
   -[[BCK-meet-semilattices]]   -[[BCK-meet-semilattices]]
-  -[[Bilattices]] 
   -[[Bilinear algebras]]   -[[Bilinear algebras]]
-  -[[Binars]], **[[Finite binars]], [[Finite binars with identity|with identity]], [[Finite binars with zero|with zero]], [[Finite binars with identity and zero|with identity and zero]]**,  
   -[[Basic logic algebras|BL-algebras]]   -[[Basic logic algebras|BL-algebras]]
-  -[[Boolean algebras]] [[wp>Boolean algebra (structure)]]+  -[[Boolean algebras]]
   -[[Boolean algebras with operators]]   -[[Boolean algebras with operators]]
   -[[Boolean groups]]   -[[Boolean groups]]
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   -[[Brouwerian algebras]]   -[[Brouwerian algebras]]
   -[[Brouwerian semilattices]]   -[[Brouwerian semilattices]]
-  -[[wp>C*-algebras]]+  -[[wp>C-star-algebras]]
   -[[Cancellative commutative monoids]]   -[[Cancellative commutative monoids]]
   -[[Cancellative commutative semigroups]]   -[[Cancellative commutative semigroups]]
   -[[Cancellative monoids]]   -[[Cancellative monoids]]
-  -[[Cancellative partial monoids]] 
   -[[Cancellative semigroups]]   -[[Cancellative semigroups]]
   -[[Cancellative residuated lattices]]   -[[Cancellative residuated lattices]]
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   -[[Closure algebras]]   -[[Closure algebras]]
   -[[Commutative BCK-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 groupoids]]
-  -[[Commutative idempotent involutive residuated lattices]] +
-  -[[Commutative integral ordered monoids]], **[[Finite commutative integral ordered monoids]]**+
   -[[Commutative inverse semigroups]]   -[[Commutative inverse semigroups]]
-  -[[Commutative involutive FL-algebras]] 
-  -[[Commutative involutive residuated posets]] 
   -[[Commutative lattice-ordered monoids]]   -[[Commutative lattice-ordered monoids]]
   -[[Commutative lattice-ordered rings]]   -[[Commutative lattice-ordered rings]]
   -[[Commutative lattice-ordered semigroups]]   -[[Commutative lattice-ordered semigroups]]
-  -[[Commutative monoids]], **[[Finite commutative monoids]], [[Finite commutative monoids with zero]]**+  -[[Commutative monoids]]
   -[[Commutative ordered monoids]]   -[[Commutative ordered monoids]]
   -[[Commutative ordered rings]]   -[[Commutative ordered rings]]
-  -[[Commutative ordered semigroups]], **[[Finite commutative ordered semigroups]]**+  -[[Commutative ordered semigroups]]
   -[[Commutative partially ordered monoids]]   -[[Commutative partially ordered monoids]]
   -[[Commutative partially ordered semigroups]]   -[[Commutative partially ordered semigroups]]
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   -[[Commutative rings]]   -[[Commutative rings]]
   -[[Commutative rings with identity]]   -[[Commutative rings with identity]]
-  -[[Commutative semigroups]], **[[Finite commutative semigroups]], [[Finite commutative semigroups with zero|with zero]]**+  -[[Commutative semigroups]]
   -[[Compact topological spaces]]   -[[Compact topological spaces]]
   -[[Boolean spaces|Compact zero-dimensional Hausdorff spaces]]   -[[Boolean spaces|Compact zero-dimensional Hausdorff spaces]]
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   -[[Directed complete partial orders|Complete partial orders]]   -[[Directed complete partial orders|Complete partial orders]]
   -[[Completely regular Hausdorff spaces]]   -[[Completely regular Hausdorff spaces]]
-  -[[wp>Completely regular semigroups]] +  -[[Completely regular semigroups]]
-  -[[Conjugative binars]]+
   -[[Continuous lattices]]   -[[Continuous lattices]]
   -[[Continuous posets]]   -[[Continuous posets]]
-  -[[Cyclic involutive FL-algebras]] 
-  -[[Cyclic involutive residuated posets]] 
   -[[Cylindric algebras]]   -[[Cylindric algebras]]
   -[[De Morgan algebras]]   -[[De Morgan algebras]]
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   -[[Distributive double p-algebras]]   -[[Distributive double p-algebras]]
   -[[Distributive dual p-algebras]]   -[[Distributive dual p-algebras]]
-  -[[Distributive involutive residuated lattices]] 
   -[[Distributive lattice expansions]]   -[[Distributive lattice expansions]]
   -[[Distributive lattices]]   -[[Distributive lattices]]
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   -[[Dunn monoids]]   -[[Dunn monoids]]
   -[[Dynamic algebras]]   -[[Dynamic algebras]]
-  -[[Effect algebras]] 
   -[[Medial groupoids|Entropic groupoids]]   -[[Medial groupoids|Entropic groupoids]]
   -[[Equivalence algebras]]   -[[Equivalence algebras]]
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   -[[FL-algebras]]   -[[FL-algebras]]
   -[[FLc-algebras]]   -[[FLc-algebras]]
-  -[[Heyting algebras|FLcw-algebras]] 
   -[[FLe-algebras]]   -[[FLe-algebras]]
-  -[[FLec-algebras]] 
-  -[[Heyting Algebras|FLecw-algebras]] 
   -[[FLew-algebras]]   -[[FLew-algebras]]
-  -[[FLi-algebras]] 
-  -[[FLo-algebras]] 
   -[[FLw-algebras]]   -[[FLw-algebras]]
   -[[Frames]]   -[[Frames]]
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   -[[Generalized BL-algebras]]   -[[Generalized BL-algebras]]
   -[[Generalized Boolean algebras]]   -[[Generalized Boolean algebras]]
-  -[[Generalized effect algebras]] 
   -[[Generalized MV-algebras]]   -[[Generalized MV-algebras]]
-  -[[Generalized orthoalgebras]] +  -[[Goedel algebras]]
-  -[[Generalized pseudo-effect algebras]] +
-  -[[Generalized separation algebras]] +
-  -[[Gödel algebras]]+
   -[[Graphs]]   -[[Graphs]]
   -[[Groupoids]]   -[[Groupoids]]
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   -[[Hilbert spaces]]   -[[Hilbert spaces]]
   -[[Hoops]]   -[[Hoops]]
-  -[[Idempotent residuated lattices]] 
-  -[[Idempotent involutive residuated lattices]] 
   -[[Idempotent semirings]]   -[[Idempotent semirings]]
   -[[Idempotent semirings with identity]]   -[[Idempotent semirings with identity]]
   -[[Idempotent semirings with identity and zero]]   -[[Idempotent semirings with identity and zero]]
   -[[Idempotent semirings with zero]]   -[[Idempotent semirings with zero]]
-  -[[Tarski algebras|Implication algebras]]+  -[[Implication algebras]]
   -[[Implicative lattices]]   -[[Implicative lattices]]
   -[[Integral domains]]   -[[Integral domains]]
-  -[[Integral involutive FL-algebras]] +  -[[Integral ordered monoids]]
-  -[[Integral ordered monoids]], **[[finite integral ordered monoids]]**+
   -[[Integral relation algebras]]   -[[Integral relation algebras]]
   -[[Integral residuated lattices]]   -[[Integral residuated lattices]]
   -[[Intuitionistic linear logic algebras]]   -[[Intuitionistic linear logic algebras]]
   -[[Inverse semigroups]]   -[[Inverse semigroups]]
-  -[[Involutive FL-algebras]] 
   -[[Involutive lattices]]   -[[Involutive lattices]]
-  -[[Involutive FL-algebras|Involutive residuated lattices]] +  -[[Involutive residuated lattices]]
-  -[[Involutive residuated posets]]+
   -[[Join-semidistributive lattices]]   -[[Join-semidistributive lattices]]
   -[[Semilattices|Join-semilattices]]   -[[Semilattices|Join-semilattices]]
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   -[[Kleene algebras]]   -[[Kleene algebras]]
   -[[Kleene lattices]]   -[[Kleene lattices]]
-  -[[Kleene logic algebras]] 
   -[[Residuated partially ordered semigroups|Lambek algebras]]   -[[Residuated partially ordered semigroups|Lambek algebras]]
-  -[[Lattice effect algebras]] 
-  -[[Lattice implication algebras]] 
   -[[Lattice-ordered groups]]   -[[Lattice-ordered groups]]
   -[[Lattice-ordered monoids]]   -[[Lattice-ordered monoids]]
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   -[[Lattices]]   -[[Lattices]]
   -[[Left cancellative semigroups]]   -[[Left cancellative semigroups]]
-  -[[Left neofield]] 
   -[[Lie algebras]]   -[[Lie algebras]]
-  -[[Commutative residuated partially ordered monoids|Lineales]] 
   -[[Goedel algebras|Linear Heyting algebras]]   -[[Goedel algebras|Linear Heyting algebras]]
   -[[Linear logic algebras]]   -[[Linear logic algebras]]
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   -[[Lukasiewicz algebras of order n]]   -[[Lukasiewicz algebras of order n]]
   -[[M-sets]]   -[[M-sets]]
-  -[[Binars|Magmas]] 
   -[[Medial groupoids]]   -[[Medial groupoids]]
   -[[Medial quasigroups]]   -[[Medial quasigroups]]
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   -[[Monadic algebras]]   -[[Monadic algebras]]
   -[[Monoidal t-norm logic algebras]]   -[[Monoidal t-norm logic algebras]]
-  -[[Monoids]], **[[Finite monoids]], [[Finite monoids with zero|with zero]]** +  -[[Monoids]]
-  -[[Monounary algebras]]+
   -[[Moufang loops]]   -[[Moufang loops]]
   -[[Moufang quasigroups]]   -[[Moufang quasigroups]]
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   -[[Multisets]]   -[[Multisets]]
   -[[MV-algebras]]   -[[MV-algebras]]
-  -[[MV-effect algebras]] 
-  -[[m-zeroids]] 
   -[[Neardistributive lattices]]   -[[Neardistributive lattices]]
   -[[Near-rings]]   -[[Near-rings]]
   -[[Near-rings with identity]]   -[[Near-rings with identity]]
   -[[Near-fields]]   -[[Near-fields]]
-  -[[Neofields]] 
   -[[Nilpotent groups]]   -[[Nilpotent groups]]
   -[[Nonassociative relation algebras]]   -[[Nonassociative relation algebras]]
   -[[Bilinear algebras|Nonassociative algebras]]   -[[Bilinear algebras|Nonassociative algebras]]
   -[[Normal bands]]   -[[Normal bands]]
-  -[[Normal valued lattice-ordered groups]] 
   -[[Normed vector spaces]]   -[[Normed vector spaces]]
   -[[Ockham algebras]]   -[[Ockham algebras]]
   -[[Order algebras]]   -[[Order algebras]]
-  -[[Ordered abelian groups]] 
   -[[Ordered fields]]   -[[Ordered fields]]
   -[[Ordered groups]]   -[[Ordered groups]]
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   -[[Ordered monoids with zero]]   -[[Ordered monoids with zero]]
   -[[Ordered rings]]   -[[Ordered rings]]
-  -[[Ordered semigroups]], **[[Finite ordered semigroups]], [[Finite ordered semigroups with zero]]** +  -[[Ordered semigroups]]
-  -[[Ordered semilattices]], **[[Finite ordered semilattices]]**+
   -[[Partially ordered sets|Ordered sets]]   -[[Partially ordered sets|Ordered sets]]
   -[[Ore domains]]   -[[Ore domains]]
-  -[[Orthoalgebras]] 
   -[[Ortholattices]]   -[[Ortholattices]]
   -[[Orthomodular lattices]]   -[[Orthomodular lattices]]
   -[[p-groups]]   -[[p-groups]]
   -[[Partial groupoids]]   -[[Partial groupoids]]
-  -[[Partial monoid]] 
   -[[Partial semigroups]]   -[[Partial semigroups]]
   -[[Partially ordered groups]]   -[[Partially ordered groups]]
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   -[[FL-algebras|Pointed residuated lattices]]   -[[FL-algebras|Pointed residuated lattices]]
   -[[Polrims]]   -[[Polrims]]
-  -[[wp>Polyadic algebras]]+  -[[Polyadic algebras]]
   -[[Partially ordered sets|Posets]]   -[[Partially ordered sets|Posets]]
   -[[Post algebras]]   -[[Post algebras]]
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   -[[Process algebras]]   -[[Process algebras]]
   -[[Pseudo basic logic algebras]]   -[[Pseudo basic logic algebras]]
-  -[[Pseudo-effect algebras]] 
   -[[Pseudo MTL-algebras]]   -[[Pseudo MTL-algebras]]
   -[[Pseudo MV-algebras]]   -[[Pseudo MV-algebras]]
   -[[Pseudocomplemented distributive lattices]]   -[[Pseudocomplemented distributive lattices]]
   -[[Pure discriminator algebras]]   -[[Pure discriminator algebras]]
-  -[[Quandles]] 
   -[[Quantales]]   -[[Quantales]]
   -[[Quasigroups]]   -[[Quasigroups]]
   -[[Quasi-implication algebras]]   -[[Quasi-implication algebras]]
-  -[[Quasi-MV-algebra]] 
   -[[Preordered sets|Quasi-ordered sets]]   -[[Preordered sets|Quasi-ordered sets]]
   -[[Quasitrivial groupoids]]   -[[Quasitrivial groupoids]]
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   -[[Residuated partially ordered monoids]]   -[[Residuated partially ordered monoids]]
   -[[Residuated partially ordered semigroups]]   -[[Residuated partially ordered semigroups]]
-  -[[Residuated partially ordered monoids|Residuated posets]] 
-  -[[Right residuated binars]] 
-  -[[Right hoops]] 
-  -[[Right quasigroups]] 
   -[[Rings]]   -[[Rings]]
   -[[Rings with identity]]   -[[Rings with identity]]
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   -[[Semiassociative relation algebras]]   -[[Semiassociative relation algebras]]
   -[[Semidistributive lattices]]   -[[Semidistributive lattices]]
-  -[[Semifields]] +  -[[Semigroups]]
-  -[[Semigroups]], **[[Finite semigroups]]**+
   -[[Monoids|Semigroups with identity]]   -[[Monoids|Semigroups with identity]]
-  -[[Semigroups with zero]], **[[Finite semigroups with zero]]** +  -[[Semigroups with zero]] 
-  -[[Semilattices]], **[[Finite semilattices]]** +  -[[Semilattices]] 
-  -[[Semilattices with identity]], **[[Finite semilattices with identity]]**+  -[[Semilattices with identity]]
   -[[Semilattices with zero]]   -[[Semilattices with zero]]
   -[[Semirings]]   -[[Semirings]]
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   -[[Semirings with identity and zero]]   -[[Semirings with identity and zero]]
   -[[Semirings with zero]]   -[[Semirings with zero]]
-  -[[Separation algebras]] 
   -[[Sequential algebras]]   -[[Sequential algebras]]
   -[[Sets]]   -[[Sets]]
   -[[Shells]]   -[[Shells]]
   -[[Division rings|Skew-fields]]   -[[Division rings|Skew-fields]]
-  -[[Skew lattices]]+  -[[Skew_lattices]]
   -[[Small categories]]   -[[Small categories]]
   -[[Sober T0-spaces]]   -[[Sober T0-spaces]]
   -[[Solvable groups]]   -[[Solvable groups]]
-  -[[Sqrt-quasi-MV-algebras]] 
   -[[Stably compact spaces]]   -[[Stably compact spaces]]
   -[[Steiner quasigroups]]   -[[Steiner quasigroups]]
   -[[Stone algebras]]   -[[Stone algebras]]
-  -[[Sugihara algebras]] 
-  -[[Sugihara monoids]] 
   -[[Symmetric relations]]   -[[Symmetric relations]]
   -[[T0-spaces]]   -[[T0-spaces]]
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   -[[wp>Topological vector spaces]]   -[[wp>Topological vector spaces]]
   -[[Torsion groups]]   -[[Torsion groups]]
-  -[[Abelian ordered groups|Totally ordered abelian groups]] 
-  -[[Ordered groups|Totally ordered groups]] 
   -[[Ordered monoids|Totally ordered monoids]]   -[[Ordered monoids|Totally ordered monoids]]
   -[[Transitive relations]]   -[[Transitive relations]]

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