Something like a field but with 3 operations?

Question

I know of Groups, and Rings, and Fields but what about tacking on a 3rd operation. Is there any use in considering some structure that consists of a field but with a 3rd operation (possibly less well behaved than the other two)?

The link in the comments is helpful, but to make this question more specific I would be interested in something that is a field under the first two operations.

A very relevant post: Why don't we study algebraic objects with more than two operations? — J. W. Perry, Nov 15 '13 at 21:21
Ahh! Thank you for that link, to make the question a bit more precise I would be interested in something that requires the set to be a field under the first two operations. — Twiltie, Nov 15 '13 at 21:22

Shaun · Accepted Answer · 2019-08-25T22:50:28.933

8

Have a look through A Course in Universal Algebra, by S. Burris & H.P. Sankappanavar. You'll find many examples of algebras with more than two operations there. It builds up a rich theory for them in fact, proving, for instance, the isomorphism theorems in full generality.

edited Aug 25 '19 at 22:50

answered Nov 15 '13 at 21:58

Shaun

44,997

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Thank you for the answer, the link is exactly what I was looking for and very interesting, I'll have to read it! – Twiltie Nov 15 '13 at 22:25
@Twiltie: You're welcome :) – Shaun Nov 15 '13 at 22:30
Thank you, @Ilmari Karonen :) – Shaun Dec 23 '13 at 12:57
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No problem. FWIW, I've asked a mod to look into how that spam edit managed to get approved in the first place. – Ilmari Karonen Dec 23 '13 at 13:00

score 2 · Answer 2 · answered Nov 15 '13 at 21:23

2

Poisson algebras come to mind. They have an associative ring structure as well as a second "multiplication" that behaves as a Lie bracket, compatible with the first multiplication.

answered Nov 15 '13 at 21:23

Sammy Black

25,273

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Thank you for the answer! That is close to what I am looking for but the operations seem a little more restricted than just some arbitrary field with an operation tacked on. If nothing else comes up in a couple of hours I will accept! – Twiltie Nov 15 '13 at 21:29

Something like a field but with 3 operations?

2 Answers2