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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.

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

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Have you read en.wikipedia.org/wiki/Algebraic_structure ? There is a very long list of examples. $\endgroup$ – Martin Brandenburg Jan 4 '14 at 22:58
  • $\begingroup$ 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. $\endgroup$ – Grigory M Jan 4 '14 at 23:04
  • $\begingroup$ 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). $\endgroup$ – Steven Stadnicki Jan 4 '14 at 23:55
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Quandles arise (allegedly) quite naturally in knot theory. They are also connected to group theory, since the conjugation operation on a group gives rise to a quandle.

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    $\begingroup$ ... 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 $\pi_1(S^3 \setminus K)$. $\endgroup$ – Sammy Black Jan 4 '14 at 23:22
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You might be interested in Universal Algebra. You could build your own algebra that way. Have a look through A Course in Universal Algebra, by S. Burris and H. P. Sankappanavar; it builds up a wonderful theory for them. Examples from this book include "squags" and "sloops".

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Groupoids seem somewhat seem somewhat more exotic than groups, but are actually a lot more natural in many ways. For example the fundamental groupoid is really more "fundamental" than the fundamental group in many ways.

Operads are also quite handy when doing algebraic topology. I recommend looking at Tom Leinster's book "Higher operads, higher categories".

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