neofields - MathStructures

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=====Neofileds=====

Abbreviation: **Nfld**
====Definition====
A \emph{neofield} is a structure $\mathbf{F}=\langle F,+,\backslash,/,0,\cdot,1,^{-1}\rangle $ of type $\langle 2,2,2,0,2,0,1\rangle $ such that


$\langle F,+,\backslash,/,0\rangle $ is a [[loop]]


$\langle F-\{0\},\cdot,1,^{-1}\rangle$ is a [[group]]


$\cdot$ distributes over $+$:  $x\cdot(y+z)=x\cdot y+x\cdot z$ and $(x+y)\cdot z=x\cdot z+y\cdot z$

==Morphisms==
Let $\mathbf{F}$ and $\mathbf{K}$ be neofields. A morphism from $\mathbf{F}$
to $\mathbf{K}$ is a function $h:F\to K$ that is a homomorphism: 

$h(x+y)=h(x)+h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$, $h(0)=0$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(1)=1$

====Examples====
Example 1: 

====Basic results====

====Properties====
^[[Classtype]]  | |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  | |
^[[Residual size]]  | |
^[[Congruence distributive]]  | |
^[[Congruence modular]]  | |
^[[Congruence n-permutable]]  | |
^[[Congruence regular]]  | |
^[[Congruence uniform]]  | |
^[[Congruence extension property]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
\end{array}$

====Subclasses====
[[Division rings]] 

====Superclasses====
[[Left neofields]] 

====References====

Paige L.J., Neofields, Duke Math. J. 16 (1949), 39--60.

[(Ln19xx>
)]

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