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.



Generalizations of Rings with multiple higher order Operators
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](http://math.berkeley.edu/~gbergman/245/). I found this to be a very nice introduction to this subject.


