Generalizations of Rings with multiple higher order Operators - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:10:18Z http://mathoverflow.net/feeds/question/25113 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25113/generalizations-of-rings-with-multiple-higher-order-operators Generalizations of Rings with multiple higher order Operators bsdz 2010-05-18T10:25:44Z 2010-05-18T23:16:43Z <p>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.</p> <p>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. </p> <p>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.</p> <p>Another possibility is that this is something covered using Category Theory; then perhaps someone could point me to primer that covers the concept described.</p> http://mathoverflow.net/questions/25113/generalizations-of-rings-with-multiple-higher-order-operators/25116#25116 Answer by Andrew Stacey for Generalizations of Rings with multiple higher order Operators Andrew Stacey 2010-05-18T11:07:13Z 2010-05-18T11:07:13Z <p>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 <a href="http://math.berkeley.edu/~gbergman/245/" rel="nofollow">here</a>. I found this to be a very nice introduction to this subject.</p> http://mathoverflow.net/questions/25113/generalizations-of-rings-with-multiple-higher-order-operators/25185#25185 Answer by David Carchedi for Generalizations of Rings with multiple higher order Operators David Carchedi 2010-05-18T23:16:43Z 2010-05-18T23:16:43Z <p>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.</p>