# What are some examples of “exotic” algebraic structures? [closed]

I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, but they seem to be way to general notions as to admit a really interesting theory.

Thus, I'm wondering whether there are also other interesting algebraic structure (here, this means mainly some set $S$ together with a bunch of functions $f_i:S^n\to S$ satisfying some laws) which behave somewhat differently, i. e. satisfy some unusual relations like $(ab)c=(ca)(cb)$ or $ba=(aa)(bb)$, but in such a way that there is a decent amount of theory about them (some kind of nontrivial classification or representation theorem would be truly fascinating).

Bonus points if these structures arise naturally in some areas of mathematics.

## closed as too broad by Grigory M, Old John, Sharkos, user7530, Joe Johnson 126Jan 4 '14 at 23:55

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• Have you read en.wikipedia.org/wiki/Algebraic_structure ? There is a very long list of examples. – Martin Brandenburg Jan 4 '14 at 22:58
• There are many examples: from coalgebras and contramodules, to Gerstenhaber and BV-algebras, to $A_\infty$- and $L_\infty$-algebras, to Moufang loops... This is really to broad. – Grigory M Jan 4 '14 at 23:04
• You might find Laver Tables (en.wikipedia.org/wiki/Laver_table) interesting - they arise in studies of large-cardinal properties and elementary embeddings of set-theoretic universes, but they're finite algebraic structures (satisfying a certain distributive rule that gives them their structure). – Steven Stadnicki Jan 4 '14 at 23:55

• ... and the slightly more general notion of rack. By labeling the stands of a knot diagram (labels continue over overpasses, but don't under underpasses), and insisting that your algebraic structure behaves under Reidemeister moves, the axioms of the rack are forced. The example of labeling each stand with a generator of a group, and relating each understrand incident at a crossing by conjugation by the overcrossing element, you get the Wirtinger presentation of the knot group $\pi_1(S^3 \setminus K)$. – Sammy Black Jan 4 '14 at 23:22