I hope this question is not too basic. I've asked various mathematicians in the past and had a good search through the Internet but with not a lot of luck.

I am interested in generalizations of Groups or Rings with more than the standard one or two operators. Perhaps one might say Sets with multiple (>2) ternary or even higher order operators.

I suspect someone may have at some point in the past proved that any generalisation can be reduced to a combination of Rings or Groups.

Another possibility is that this is something covered using Category Theory; then perhaps someone could point me to primer that covers the concept described.

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    $\begingroup$ Have you any examples of the structures you have in mind? Then you might be able to extract the salient features of such examples and suggest interesting axiom systems for the operations they have. Recall that ring theory arose from considering the common features of structures such as the integers, rationals, reals Gaussian integers and so forth. (The general theory of structures with $k$-ary operations is known as "universal algebra"). $\endgroup$ May 18, 2010 at 10:30
  • $\begingroup$ I have some scribbled down but haven't rigorously tested them enough to post here. Also not sure on a standard way to TeX some of them. I shall investigate Universal Algebras. Thank you. $\endgroup$ May 18, 2010 at 11:16
  • $\begingroup$ I notice that this has a vote to close but no comment as to why. I thought I'd comment on why I didn't vote to close, since I did consider it. I thought it possible that a satisfactory answer to this question would be a reference, so I classed this question as a "request for reference". For such questions, I have a much higher tolerance because for someone who knows, they are really easy to answer. I also know that if you don't know where to start looking, it can be very frustrating. If anyone would like to comment on my opinion, start a thread on meta (I shan't look here again). $\endgroup$ May 19, 2010 at 7:25

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As Robin says in his comment, the general framework of this is "universal algebra" or "Lawvere theories". Given the phrasing of the question, I shall refrain from directing you to the relevant nLab pages but instead point you to George Bergman's "An Invitation to General Algebra and Universal Constructions" which is available from his homepage here. I found this to be a very nice introduction to this subject.

  • $\begingroup$ Thank you for the pointers. I will revisit nLab and shall investigate the link :) $\endgroup$ May 18, 2010 at 11:18
  • $\begingroup$ One of the nice things about George Bergman's notes is that you can learn the category theory as you go along. $\endgroup$ May 18, 2010 at 11:38

A ring is a monoid internal to abelian groups. On the other hand, a monoid is a one-object category. Therefore, a natural generalization (horizontal categorification) of a ring is an operad internal to abelian groups. Whether or not much formal ring theory has been studied with these objects, I do not know. I would be interested if you find anything out however.

  • $\begingroup$ Thank you for suggesting a Category aspect. If I discover anything that might be of interest to you (operad internal to a.g.s) I will post a message. Though, it might be a very long time before I get there. $\endgroup$ May 19, 2010 at 13:54

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