Slightly weakened / altered concepts of a field 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!
 A: One weakening of the field axioms that I've heard of several times is that of a near field.  A near field satisfies the axioms of a division ring, except that it does not have distributivity of right multiplication.
Here's a fun example: look at the quaternions a+bi+cj+dk with a,b,c,d integers.  Look at the element 1+i+j.  Quotienting out by the left ideal generated by this element gives a 9 element additive group.  A short calculation shows that the elements $0, \pm 1, \pm i, \pm j, \pm k$ give a basis for this group.  Hence the non-zero elements of this additive group have a multiplicative structure!  It's easy to check that this is left distributive but not right distributive.
One reason that near fields are useful is that doubly transitive Frobenius groups are precisely semidirect products of the form $k^+ \rtimes k^\times$ for k a finite near field.  
For two papers where near fields come up see this old paper of mine on large representations of finite groups or a paper that came out today on quantum doubles of finite groups by Beigi, Shor, Whalen
A: Nearfields and semifields and such things appear when you try to coordinatize (or whatever the right word is) projective planes (in the combinatorial sense).
A: The meadow (as defined in the question, and in the paper linked) is an "equational theory".  
A meadow is a set $A$ together with operations $0,1,+,-,\cdot,{}^{-1}$ such that $(A,0,1,+,-,\cdot)$ is a commutative ring with unit, and identities
$$
(x^{-1})^{-1} = x
\\
x\cdot(x\cdot x^{-1}) = x
$$
hold for all $x \in A$.
As with all equational theories, this tells us what are the morphisms, subalgebras, ideals, products, quotients, and so on.  So: if $A, B$ are meadows, then a map $f : A \to B$ is a homomorphism iff
$$
f(0)=0\\
f(1)=1\\
f(x+y)=f(x)+f(y)
\\
f(-x)=-f(x)\\
f(xy) = f(x)f(y)
\\
f(x^{-1})=f(x)^{-1}
$$
(Some of these follow from the others, but abstractly you just say it preserves all the operations.)
A submeadow of a meadow $A$ is a subset $B \subseteq A$ such that
if $x,y \in B$, then
$$
0, 1, x+y, -x, x\cdot y, x^{-1} \in B
$$  
An important example of a meadow is a field, with the usual partial operation $x^{-1}$ enhanced to a total operation by defining $0^{-1}=0$.  This is called a zero totalized field.
More interesting examples are products of fields.  For example $\mathbb{Md}_6 = \mathbb F_2 \times \mathbb F_3$.
Theorem: Any meadow is (up to isomorphism) a submeadow of a product of zero totalized fields.
As Robin Chapman noted (quoted in the paper mentioned): Take a meadow and forget the inverse operation, and you have a von Neumann regular ring; start with a von Neumann regular ring with unit, there is a unique way to define the inverse making it a meadow.
