MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 4 added tags, asked for examples, added definitions i found

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

meadow, meadowswhich (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 neofields$x·(x·x^{−1}) = x$.

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). I've heard But only having these definitionsfor the first two, but I'm I still not sure 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).

Can

So, can anyone...

  • provide explicit better/more explanatory definitions for these structures(and, , 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 (book, a book studying them, or paper where they were first defined, etc.)

  • 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

    		•
    show/hide this revision's text 3 added 1 characters in body

    I've heard of at least three slight modifications of the standard concept of field: wheels, meadows, and neofields (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). I've seen heard definitions for the first two, but I'm still not sure why any of these are natural things to look at or what theorems there are to be had about them.

    Can anyone...

    1. provide explicit definitions for these structures (and, as a proper category theory student, their morphisms)

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

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

    4. 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 if you have even more things to say about them - even better!
    show/hide this revision's text 2 deleted 40 characters in body

    I've heard of at least three slight modifications of the standard concept of field: wheels, meadows, and neofields (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). However, I have only a vague notion of what the actual I've seen definitions of these structures arefor the first two, and no idea but I'm still not sure why any of these are natural things to look at or what theorems there are to be had about them.

    Can anyone...

    1. provide explicit definitions for any of these structures (and, as a proper category theory student, their morphisms)

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

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

    4. 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 if you have even more things to say about them - even better!
    show/hide this revision's text 1