I've *heard of* at least three slight modifications of the standard concept of field:

**meadow**, which (according to this paper) is a commutative ring with unit equipped with a total unary operation $x^{−1}$,
named inverse, that satisfies these additional equations:
$(x^{−1})^{−1} = x$ and $x·(x·x^{−1}) = x$.

**wheel** - I heard of these in conversation, so I'm unsure of their exact definition. I believe they have a unary "inverse" operation like meadows, but I assume something is different about them.

**neofield**, which (according to this paper) appear to be fields, without the associativity of addition.

(I'm not going to count $\mathbb{F}_1$, I don't think it's relevant here - though I may be totally off about this). But only having these definitions, I still feel unsatisfied with the concepts. I don't feel like I understand what's going on with them, I don't know why any of these are natural things to look at, or what important theorems there are to be had about them (I mean, other than the ones in the papers I referenced, which I would hopefully understand after getting a better grounding).

So, can anyone...

provide better/more explanatory definitions for these structures, and also - as a proper category theory student - their

*morphisms*(the papers I linked to don't state that, I think)provide instructive examples of each structure (i.e. examples which are not also fields, demonstrating the differences)

provide whatever are considered to be "standard" references for any of these structures (a book studying them, or paper where they were first defined, etc.)

explain why we should look at these structures (I mean, beyond just curiosity about them) - where do they naturally arise, if anywhere?

explain which classic concepts/theorems about fields carry over to each structure (do they have a notion of algebraic elements? is there a Galois-like theory for them? etc.) and which don't

and if you have even more things to say about them - even better!

abspart of an arxiv paper rather than to an actual PDF file stored in a cache. – Loop Space Mar 2 '10 at 15:09