Catalogue of Algebraic Systems

• Bilinear algebras/ Algebras / Associative algebras
• Division algebras
• Boolean algebras
• Cylindric algebras
• Fields
• Groupoids
• Entropic groupoids / Medial groupoids
• Groups
• Abelian groups
• Nilpotent groups
• Solvable groups
• Inverse semigroups
• Jordan algebras
• Kleene algebras
• Lattices
• Distributive lattices
• Modular lattices
• Orthomodular lattices
• Lie algebras
• Loops
• Inverse property loops / IP-loops
• Moufang loops
• Steiner loops / Sloops / TS-loops
• Modules
• Monoids
• Commutative monoids
• Near-fields / Nearfields
• Near-rings / Nearrings
• Post algebras
• Quasigroups
• Inverse property quasigroups / IP-quasigroups
• Medial quasigroups
• Moufang quasigroups
• Steiner quasigroups / Squags
• TS-quasigroups
• Rings
• Commutative rings
• Division rings / Skew fields
• Regular rings
• Rings with identity / Rings with 1 / Unital rings
• Semigroups
• Commutative semigroups
• Regular semigroups
• Vector spaces




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Additional Links

Since the above catalogue was compiled much more information on algebraic matters has appeared on the web. Below are links to some of these webpages. I have just begun collecting them . If you know of others that would be appropriate to add please let me know. As of now they are listed in no particular order. Organization may come later. --W. Edwin Clark eclark@math.usf.edu
 
• Geoffrey Dixon's Technical Pages on Division Algebras (especially, Quaternions and Octonions), Lie Algebras, Lie Groups, Spinors and Lattices
• To see how much we have to add, browse: 1991 Mathematics Subject Classification
• Atlas of Finite Group Representations. by Robert Wilson, et al.
• The Number of Groups of Order n where n is at most 1000 and other Combinatorial Data. by Gordon Royle
• GAP , Groups, Algorithms and Programming, is a FREE system for computational discrete algebra with particular emphasis on computational group theory. GAP includes various databases including all groups of order up to 1,000, excluding 512 and 768.
• Perhaps the most general algebraic structure (outside of category theory) is the universal algebra. For a brief introduction and some history see: Hermann Grassmann and the Prehistory of Universal Algebra by Desmond Fearnley-Sander
• Three element groupoids. by Stanley Burris
• Three element groupoids with unary operation. by Stanley Burris
• Dave Rusin's Algebraic Areas of Mathematics from his Mathematical Atlas This itself contains quite a catalogue of algebraic systems.
• Eric Weisstein's Encyclopedia of Mathematics Many algebraic systems are described here.
• Modern Algebra Topics from the Math Forum's Internet Mathematics Library.
• Algebraic Systems This is essentially a textbook on algebraic systems put together by Christer Blomqvist. Compared to Weisstein's site above, this is more of a tutorial. So take a look if you want more discussion. It is written in English, but includes A very brief English to Swedish dictionary .
• Groupoid Homepage These groupoids are small categories where all morphism are isomorphisms and are not to be confused with the other groupoids which are sets with a single binary operation. See Ronald Brown's discussion of groupoids in the categorical sense.
• Applied and Computational Category Theory This includes a brief history and description of category theory.
• Higher Dimensional Group Theory By Ronald Brown.
• E. Lee Lady's Notes This includes Notes on Homological Algegra, A Course in Abstract Algebra, and Torsion Free Modules over Dedekind Domains.
• The Net Advance of Physics This is a list of links similar to the one you are in now of topics of an algebraic nature--many, but not all, related to physics. The list is compiled and managed by Norman Redington.
• Zero Divisor Structure in Real Algebras by Steve Finch.
• Sketch of the history of hypercomplex numbers Note that hypercomplex numbers are better known as associative algebras .
• Logic and Philosophy of Mathematics Section of Stanford's Encyclopedia of Philosophy This includes entries for many interesting topics including category theory and the home page for the axiom of choice .
• Semigroup Mailing List A mailing list to go to for technical questions about semigroups.
• Small Semigroups Reference to a paper on the number of semigroups of orders up to 8 by S. Satoh, et al.
• Abstract Algebra On Line This site contains many of the definitions and theorems from abstract algebra. It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course. It is based on the following books: Abstract Algebra Second Edition, by John A. Beachy and William D. Blair, and Abstract Algebra II, by John A. Beachy.
• The Rubik's cube group ( which has order 43252003274489856000) from the homepage of W. D. Joyner which links to Joyner's Permutation Puzzle Page and other algebraic matters of interest.
• Rubik's Cube Page A LOT of Rubik's cube information by Michael Reid.;
• Rubik's Cube Solutions by Josef "Gloom" Jelinek and Dr. Hana M. Bizek.
• Maple Worksheets for Abstract Algebra This is a collection of Maple worksheets by Fr Mike May, S.J. The worksheets are mainly concerned with field theory--Galois theory, finite fields, etc.
• Diamond Theory by Steven H. Cullinane. See also The Diamond 16 Puzzle also by Steven Cullinane. The puzzle displays the affine group on the 4-space over GF(2) as generated by row/column/quadrant permutations of a 4x4 array.
• Exploring Abstract Algebra with Mathematica by Al Hibbard  and Ken Levasseur. This is a collection of Mathematica packages that enable one to work with concepts from abstract algebra (including groups, rings, fields, and morphisms).