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.

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 and would like some information added, please email Peter Jipsen (jipsen@chapman.edu). [Account creation unfortunately had to be disabled.]

New: Boldface links below display lists of finite algebras in a particular class (calculated locally in the browser).

Acknowledgements | Notation and terminology | Properties | Tools | Online books and lecture notes

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

Algebras | Logics | Syntax | Terms | Equations | Horn formulas | Universal formulas | First-order formulas

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

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