Abbreviation: Nfld

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 $(x+y)\cdot z=x\cdot z+y\cdot z$

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:

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

Example 1:

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

Division rings

Left neofields

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