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Mathematical Structures

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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 first-order classes of relational structures and, more particularly, varieties of universal algebras. If you are familiar with some of these classes of algebras, feel free to add some relevant information and references by using the edit link at the bottom of the respective page. A phrase followed by a blue '?' is a link to a page that still needs to be filled in. Adding and formatting pages is really simple. Here are some basic [text formatting rules] (see also Wikipedia[how to edit a page], although not all the features mentioned there are available here). Mathematical notation is delimited by dollar signs, and uses a very basic subset of LaTeX (essentially those characters and symbols that can be displayed by most browsers). You can use this sandbox page to try out editing (without worrying about deleting useful information).

New: There is now an experimental XHTML+MathML version of these pages at http://math.chapman.edu/cgi-bin/structuresxml.pl (go to [Display MathML on these pages] to check if your browser can handle MathML formulas).

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Please use this template for new pages and follow the format of other pages that have already been partially filled out. If possible, the value of a property should be justified by a brief reason or a reference to a publication where the result can be found. Bibtex references (e.g. from MathSciNet) can be automatically translated to the bibxml format using the [bibtex2xml converter].

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

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

Classes of ordered algebraic structures | Partial algebras | Many-sorted algebras | Logics

Alphabetical list of all classes

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