# Differences

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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 add or correct information.// | + | These pages 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 | Initially the main content concerns mostly first-order classes of relational | ||

structures and, more particularly, | structures and, more particularly, | ||

- | If you are familiar with some of these classes of structures, feel | + | 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 | 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 | + | link on the respective page. Pages are written in DokuWiki syntax, but mathematical notation is in standard LaTeX. |

- | you don't know LaTeX you should probably not edit these pages -- the previous non-LaTeX | + | The previous |

- | version of these pages is still available at http://). | + | versions of these pages are still available at http://and http://. |

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.*/ |

- | This front page can also be edited to add further classes of structures. A criterion | + | **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 | 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]] | ||

- | [[Acknowledgements]] | [[Notation and terminology]] | [[Properties]] | | + | [[Varieties]] | [[Quasivarieties]] | [[Universal classes]] | [[First-order classes]] | [[Second-order classes]] |

- | [[Tools]] | [[Online books and lecture notes]] | + | |

- | [[Varieties]] | [[Quasivarieties]] | [[Universal classes]] | [[First-order classes]] | | + | [[Algebras]] | [[Logics]] | [[Syntax]] | [[Terms]] | [[Equations]] | [[Horn formulas]] | [[Universal formulas]] | [[First-order formulas]] /*| [[Theories]]*/ |

- | [[Second-order classes]] | [[Logics]] | + | |

=== Alphabetical list of all classes === | === Alphabetical list of all classes === | ||

- | -[[Abelian groups]] | + | -[[Abelian groups]] nbsp [[wp>]] |

-[[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]] | + | -[[Action algebras]] nbsp [[wp>]] |

-[[Action lattices]] | -[[Action lattices]] | ||

-[[Algebraic lattices]] | -[[Algebraic lattices]] | ||

- | -[[Algebraic posets]] | + | -[[Algebraic posets]] nbsp [[wp>]] |

-[[Algebraic semilattices]] | -[[Algebraic semilattices]] | ||

- | -[[Allegories]] | + | -[[Allegories]] nbsp [[wp>]] |

-[[Almost distributive lattices]] | -[[Almost distributive lattices]] | ||

- | -[[Associative algebras]] | + | -[[Associative algebras]] nbsp [[wp>]] |

- | -[[Banach spaces]] | + | -[[Banach spaces]] nbsp [[wp>]] |

- | -[[Bands]] | + | -[[Bands]] nbsp [[wp> |

-[[Basic logic algebras]] | -[[Basic logic algebras]] | ||

- | -[[BCI-algebras]] | + | -[[BCI-algebras]] nbsp [[wp>BCI algebra]] |

- | -[[BCK-algebras]] | + | -[[BCK-algebras]] nbsp [[wp>BCK algebra]] |

-[[BCK-join-semilattices]] | -[[BCK-join-semilattices]] | ||

-[[BCK-lattices]] | -[[BCK-lattices]] | ||

-[[BCK-meet-semilattices]] | -[[BCK-meet-semilattices]] | ||

+ | -[[Bilattices]] | ||

-[[Bilinear algebras]] | -[[Bilinear algebras]] | ||

+ | -[[Binars]], | ||

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

- | -[[Boolean algebras]] | + | -[[Boolean algebras]] nbsp [[wp>]] |

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

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

Line 68: | Line 73: | ||

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

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

- | -[[wp>C-star-algebras]] | + | -[[wp>C*-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]] | ||

Line 80: | Line 86: | ||

-[[Closure algebras]] | -[[Closure algebras]] | ||

-[[Commutative BCK-algebras]] | -[[Commutative BCK-algebras]] | ||

- | -[[Commutative groupoids]] | + | -[[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 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]] | + | -[[Commutative monoids]], **[[Finite commutative monoids]], [[Finite commutative monoids with zero]]** |

-[[Commutative ordered monoids]] | -[[Commutative ordered monoids]] | ||

-[[Commutative ordered rings]] | -[[Commutative ordered rings]] | ||

- | -[[Commutative ordered semigroups]] | + | -[[Commutative ordered semigroups]], **[[Finite commutative ordered semigroups]]** |

-[[Commutative partially ordered monoids]] | -[[Commutative partially ordered monoids]] | ||

-[[Commutative partially ordered semigroups]] | -[[Commutative partially ordered semigroups]] | ||

Line 98: | Line 108: | ||

-[[Commutative rings]] | -[[Commutative rings]] | ||

-[[Commutative rings with identity]] | -[[Commutative rings with identity]] | ||

- | -[[Commutative semigroups]] | + | -[[Commutative semigroups]], **[[Finite commutative semigroups]], |

-[[Compact topological spaces]] | -[[Compact topological spaces]] | ||

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

Line 109: | Line 119: | ||

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

-[[Completely regular Hausdorff spaces]] | -[[Completely regular Hausdorff spaces]] | ||

- | -[[Completely regular semigroups]] | + | -[[wp>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]] | ||

Line 126: | Line 139: | ||

-[[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]] | ||

Line 137: | Line 151: | ||

-[[Dunn monoids]] | -[[Dunn monoids]] | ||

-[[Dynamic algebras]] | -[[Dynamic algebras]] | ||

+ | -[[Effect algebras]] | ||

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

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

Line 145: | Line 160: | ||

-[[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]] | ||

Line 153: | Line 173: | ||

-[[Generalized BL-algebras]] | -[[Generalized BL-algebras]] | ||

-[[Generalized Boolean algebras]] | -[[Generalized Boolean algebras]] | ||

+ | -[[Generalized effect algebras]] | ||

-[[Generalized MV-algebras]] | -[[Generalized MV-algebras]] | ||

- | -[[Goedel algebras]] | + | -[[Generalized orthoalgebras]] |

+ | -[[Generalized pseudo-effect algebras]] | ||

+ | -[[Generalized separation algebras]] | ||

+ | -[[Gödel algebras]] | ||

-[[Graphs]] | -[[Graphs]] | ||

-[[Groupoids]] | -[[Groupoids]] | ||

Line 163: | Line 187: | ||

-[[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]] | ||

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

-[[Implicative lattices]] | -[[Implicative lattices]] | ||

-[[Integral domains]] | -[[Integral domains]] | ||

- | -[[Integral ordered monoids]] | + | -[[Integral involutive FL-algebras]] |

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

+ | -[[Involutive residuated posets]] | ||

-[[Join-semidistributive lattices]] | -[[Join-semidistributive lattices]] | ||

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

Line 182: | Line 211: | ||

-[[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]] | ||

Line 189: | Line 221: | ||

-[[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]] | ||

Line 198: | Line 232: | ||

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

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

+ | -[[Binars|Magmas]] | ||

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

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

Line 209: | Line 244: | ||

-[[Monadic algebras]] | -[[Monadic algebras]] | ||

-[[Monoidal t-norm logic algebras]] | -[[Monoidal t-norm logic algebras]] | ||

- | -[[Monoids]] | + | -[[Monoids]], **[[Finite monoids]], [[Finite monoids with zero|with zero]]** |

+ | -[[Monounary algebras]] | ||

-[[Moufang loops]] | -[[Moufang loops]] | ||

-[[Moufang quasigroups]] | -[[Moufang quasigroups]] | ||

Line 217: | Line 253: | ||

-[[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]] | ||

Line 233: | Line 274: | ||

-[[Ordered monoids with zero]] | -[[Ordered monoids with zero]] | ||

-[[Ordered rings]] | -[[Ordered rings]] | ||

- | -[[Ordered semigroups]] | + | -[[Ordered semigroups]], **[[Finite ordered semigroups]], |

+ | -[[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 249: | Line 293: | ||

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

-[[Polrims]] | -[[Polrims]] | ||

- | -[[Polyadic algebras]] | + | -[[wp>Polyadic algebras]] |

-[[Partially ordered sets|Posets]] | -[[Partially ordered sets|Posets]] | ||

-[[Post algebras]] | -[[Post algebras]] | ||

Line 257: | Line 301: | ||

-[[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]] | ||

Line 282: | Line 329: | ||

-[[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]] | ||

Line 287: | Line 338: | ||

-[[Semiassociative relation algebras]] | -[[Semiassociative relation algebras]] | ||

-[[Semidistributive lattices]] | -[[Semidistributive lattices]] | ||

- | -[[Semigroups]] | + | -[[Semifields]] |

+ | -[[Semigroups]], **[[Finite semigroups]]** | ||

-[[Monoids|Semigroups with identity]] | -[[Monoids|Semigroups with identity]] | ||

- | -[[Semigroups with zero]] | + | -[[Semigroups with zero]], **[[Finite semigroups with zero]]** |

- | -[[Semilattices]] | + | -[[Semilattices]], **[[Finite semilattices]]** |

- | -[[Semilattices with identity]] | + | -[[Semilattices with identity]], **[[Finite semilattices with identity]]** |

-[[Semilattices with zero]] | -[[Semilattices with zero]] | ||

-[[Semirings]] | -[[Semirings]] | ||

Line 297: | Line 349: | ||

-[[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 319: | Line 375: | ||

-[[wp> | -[[wp> | ||

-[[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]] |