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An online database of Mathematical Structures

This website is currently under development.

The previous version (using LaTeX/PDF) is available at http://math.chapman.edu/cgi-bin/structures.

The first version (on a usemod wiki) is available at http://math.chapman.edu/cgi-bin/structures.pl.

The webpages collected here (will) 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. If you decide to add content or fix a mathematical statement (rather than just fixing a typo) it is suggested that you create an account and log in on this site.

Initially the main content concerns mostly first-order classes of relational structures and, more particularly, varieties of universal algebras. 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 a subset of standard LaTeX (if you don't know LaTeX you should probably not edit these pages). 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

Theorem prover input files

Varieties | Quasivarieties | Universal classes | First-order classes | Second-order classes | Logics

Alphabetical list of all classes

  1. Abelian groups
  2. Abelian lattice-ordered groups
  3. Abelian ordered groups
  4. Abelian p-groups
  5. Abelian partially ordered groups
  6. Action algebras
  7. Action lattices
  8. Algebraic lattices
  9. Algebraic posets
  10. Algebraic semilattices
  11. Allegories
  12. Almost distributive lattices
  13. Associative algebras
  14. Banach spaces
  15. Bands
  16. Basic logic algebras
  17. BCI-algebras
  18. BCK-algebras
  19. BCK-join-semilattices
  20. BCK-lattices
  21. BCK-meet-semilattices
  22. Bilinear algebras
  23. BL-algebras
  24. Boolean algebras
  25. Boolean algebras with operators
  26. Boolean groups
  27. Boolean lattices
  28. Boolean modules over a relation algebra
  29. Boolean monoids
  30. Boolean rings
  31. Boolean semigroups
  32. Boolean semilattices
  33. Boolean spaces
  34. Bounded distributive lattices
  35. Bounded lattices
  36. Bounded residuated lattices
  37. Brouwerian algebras
  38. Brouwerian semilattices
  39. C-star-algebras
  40. Cancellative commutative monoids
  41. Cancellative commutative semigroups
  42. Cancellative monoids
  43. Cancellative semigroups
  44. Cancellative residuated lattices
  45. Categories
  46. Chains
  47. Clifford semigroups
  48. Clifford algebras
  49. Closure algebras
  50. Commutative BCK-algebras
  51. Commutative groupoids
  52. Commutative inverse semigroups
  53. Commutative lattice-ordered monoids
  54. Commutative lattice-ordered rings
  55. Commutative lattice-ordered semigroups
  56. Commutative monoids
  57. Commutative ordered monoids
  58. Commutative ordered rings
  59. Commutative ordered semigroups
  60. Commutative partially ordered monoids
  61. Commutative partially ordered semigroups
  62. Commutative regular rings
  63. Commutative residuated lattice-ordered semigroups
  64. Commutative residuated lattices
  65. Commutative residuated partially ordered monoids
  66. Commutative residuated partially ordered semigroups
  67. Commutative rings
  68. Commutative rings with identity
  69. Commutative semigroups
  70. Compact topological spaces
  71. Compact zero-dimensional Hausdorff spaces
  72. Complemented lattices
  73. Complemented distributive lattices
  74. Complemented modular lattices
  75. Complete distributive lattices
  76. Complete lattices
  77. Complete semilattices
  78. Complete partial orders
  79. Completely regular Hausdorff spaces
  80. Completely regular semigroups
  81. Continuous lattices
  82. Continuous posets
  83. Cylindric algebras
  84. De Morgan algebras
  85. De Morgan monoids
  86. Dedekind categories
  87. Dedekind domains
  88. Dense linear orders
  89. Digraph algebras
  90. Directed complete partial orders
  91. Directed partial orders
  92. Directed graphs
  93. Directoids
  94. Distributive allegories
  95. Distributive double p-algebras
  96. Distributive dual p-algebras
  97. Distributive lattice expansions
  98. Distributive lattices
  99. Distributive lattices with operators
  100. Distributive lattice ordered semigroups
  101. Distributive p-algebras
  102. Distributive residuated lattices
  103. Division algebras
  104. Division rings
  105. Double Stone algebras
  106. Dunn monoids
  107. Dynamic algebras
  108. Entropic groupoids
  109. Equivalence algebras
  110. Equivalence relations
  111. Euclidean domains
  112. f-rings
  113. Fields
  114. FL-algebras
  115. FLc-algebras
  116. FLe-algebras
  117. FLew-algebras
  118. FLw-algebras
  119. Frames
  120. Function rings
  121. G-sets
  122. Generalized BL-algebras
  123. Generalized Boolean algebras
  124. Generalized MV-algebras
  125. Goedel algebras
  126. Graphs
  127. Groupoids
  128. Groups
  129. Hausdorff spaces
  130. Heyting algebras
  131. Hilbert algebras
  132. Hilbert spaces
  133. Hoops
  134. Idempotent semirings
  135. Idempotent semirings with identity
  136. Idempotent semirings with identity and zero
  137. Idempotent semirings with zero
  138. Implication algebras
  139. Implicative lattices
  140. Integral domains
  141. Integral ordered monoids
  142. Integral relation algebras
  143. Integral residuated lattices
  144. Intuitionistic linear logic algebras
  145. Inverse semigroups
  146. Involutive lattices
  147. Involutive monoids
  148. Involutive residuated lattices
  149. Join-semidistributive lattices
  150. Join-semilattices
  151. Jordan algebras
  152. Kleene algebras
  153. Kleene lattices
  154. Lambek algebras
  155. Lattice-ordered groups
  156. Lattice-ordered monoids
  157. Lattice-ordered rings
  158. Lattice-ordered semigroups
  159. Lattices
  160. Left cancellative semigroups
  161. Lie algebras
  162. Linear Heyting algebras
  163. Linear logic algebras
  164. Linear orders
  165. Locales
  166. Locally compact topological spaces
  167. Loops
  168. Lukasiewicz algebras of order n
  169. M-sets
  170. Medial groupoids
  171. Medial quasigroups
  172. Meet-semidistributive lattices
  173. Meet-semilattices
  174. Metric spaces
  175. Modal algebras
  176. Modular lattices
  177. Modular ortholattices
  178. Modules over a ring
  179. Monadic algebras
  180. Monoidal t-norm logic algebras
  181. Monoids
  182. Moufang loops
  183. Moufang quasigroups
  184. Multiplicative additive linear logic algebras
  185. Multiplicative lattices
  186. Multiplicative semilattices
  187. Multisets
  188. MV-algebras
  189. Neardistributive lattices
  190. Near-rings
  191. Near-rings with identity
  192. Near-fields
  193. Nilpotent groups
  194. Nonassociative relation algebras
  195. Nonassociative algebras
  196. Normal bands
  197. Normed vector spaces
  198. Ockham algebras
  199. Order algebras
  200. Ordered fields
  201. Ordered groups
  202. Ordered monoids
  203. Ordered monoids with zero
  204. Ordered rings
  205. Ordered semigroups
  206. Ordered sets
  207. Ore domains
  208. Ortholattices
  209. Orthomodular lattices
  210. p-groups
  211. Partial groupoids
  212. Partially ordered groups
  213. Partially ordered monoids
  214. Partially ordered semigroups
  215. Partially ordered sets
  216. Peirce algebras
  217. Pocrims
  218. Pointed residuated lattices
  219. Polrims
  220. Polyadic algebras
  221. Posets
  222. Post algebras
  223. Preordered sets
  224. Priestley spaces
  225. Principal Ideal Domains
  226. Process algebras
  227. Pseudo basic logic algebras
  228. Pseudo MTL-algebras
  229. Pseudo MV-algebras
  230. Pseudocomplemented distributive lattices
  231. Pure discriminator algebras
  232. Quantales
  233. Quasigroups
  234. Quasi-implication algebras
  235. Quasi-ordered sets
  236. Quasitrivial groupoids
  237. Rectangular bands
  238. Reflexive relations
  239. Regular rings
  240. Regular semigroups
  241. Relation algebras
  242. Relative Stone algebras
  243. Relativized relation algebras
  244. Representable cylindric algebras
  245. Representable lattice-ordered groups
  246. Representable relation algebras
  247. Representable residuated lattices
  248. Residuated idempotent semirings
  249. Residuated lattice-ordered semigroups
  250. Residuated lattices
  251. Residuated partially ordered monoids
  252. Residuated partially ordered semigroups
  253. Rings
  254. Rings with identity
  255. Schroeder categories
  256. Semiassociative relation algebras
  257. Semidistributive lattices
  258. Semigroups
  259. Semigroups with identity
  260. Semigroups with zero
  261. Semilattices
  262. Semilattices with identity
  263. Semilattices with zero
  264. Semirings
  265. Semirings with identity
  266. Semirings with identity and zero
  267. Semirings with zero
  268. Sequential algebras
  269. Sets
  270. Shells
  271. Skew-fields
  272. Small categories
  273. Sober T0-spaces
  274. Solvable groups
  275. Stably compact spaces
  276. Steiner quasigroups
  277. Stone algebras
  278. Symmetric relations
  279. T0-spaces
  280. T1-spaces
  281. T2-spaces
  282. Tarski algebras
  283. Tense algebras
  284. Temporal algebras
  285. Topological groups
  286. Topological spaces
  287. Topological vector spaces
  288. Torsion groups
  289. Totally ordered monoids
  290. Transitive relations
  291. Trees
  292. Tournaments
  293. Unary algebras
  294. Unique factorization domains
  295. Unital rings
  296. Vector spaces
  297. Wajsberg algebras
  298. Wajsberg hoops
  299. Weakly associative lattices
  300. Weakly associative relation algebras
  301. Weakly representable relation algebras

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