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Neofileds

Abbreviation: Nfld

Definition

A 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

$\langle F\backslash\{0\},\cdot,1,^{-1}\rangle$ 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

Finite members

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

Subclasses

Superclasses

References

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