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.

examplesof 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"). $\endgroup$didn'tvote 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). $\endgroup$