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start [2021/02/22 20:15]
jipsen created 		start [2021/02/22 21:11]
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 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 email additions or corrections to Peter Jipsen add or correct information.*/ +//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.
-/*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.] /*feel+If you are familiar with some of these classes of structures, feel
 free to add some relevant information and references by using the edit 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.  +link on the respective page. Most pages are written in standard LaTeX (if 
-The previous +you don't know LaTeX you should probably not edit these pages -- the previous non-LaTeX  
-versions of these pages are still available at http://math.chapman.edu/cgi-bin/structures.pl and http://math.chapman.edu/cgi-bin/structures .+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 You can use this [[Sandbox]] page to try out editing (without worrying about deleting
-useful information).*/+useful information).
  
-/*[[Suggestions or comments]]? Just follow the link and edit the page.*/+[[Suggestions or comments]]? Just follow the link and edit the page.
  
-**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 
- +
-/*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  for inclusion in this list is that there should be some journal publication or book 
-where the class has been named and defined.*/ +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]] +[[Acknowledgements]] | [[Notation and terminology]] | [[Properties]] |  
 +[[Tools]] | [[Online books and lecture notes]]
  
-[[Algebras]] | [[Logics]] | [[Syntax]] | [[Terms]] | [[Equations]] | [[Horn formulas]] | [[Universal formulas]] | [[First-order formulas]] /*| [[Theories]]*/+[[Varieties]] | [[Quasivarieties]] | [[Universal classes]] | [[First-order classes]] |  
 +[[Second-order classes]] | [[Logics]]
  
 === Alphabetical list of all classes === === Alphabetical list of all classes ===
-  -[[Abelian groups]] nbsp [[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]] nbsp [[wp>Action algebra]]+  -[[Action algebras]]
   -[[Action lattices]]   -[[Action lattices]]
   -[[Algebraic lattices]]   -[[Algebraic lattices]]
-  -[[Algebraic posets]] nbsp [[wp>Algebraic poset]]+  -[[Algebraic posets]]
   -[[Algebraic semilattices]]   -[[Algebraic semilattices]]
-  -[[Allegories]] nbsp [[wp>Allegory (category theory)]]+  -[[Allegories]]
   -[[Almost distributive lattices]]   -[[Almost distributive lattices]]
-  -[[Associative algebras]] nbsp [[wp>Associative algebra]] +  -[[Associative algebras]] 
-  -[[Banach spaces]] nbsp [[wp>Banach space]] +  -[[Banach spaces]] 
-  -[[Bands]] nbsp [[wp>Band (mathematics)]], **[[Finite bands]]**+  -[[Bands]]
   -[[Basic logic algebras]]   -[[Basic logic algebras]]
-  -[[BCI-algebras]] nbsp [[wp>BCI algebra]] +  -[[BCI-algebras]] 
-  -[[BCK-algebras]] nbsp [[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]] nbsp [[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]]
Line 253: Line 217:
   -[[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]]
Line 293: Line 249:
   -[[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]]
Line 375: Line 319:
   -[[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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