Table of Contents
Near-fields
Abbreviation: NFld
Definition
A \emph{near-field} is a near-rings with identity $\mathbf{N}=\langle N,+,-,0,\cdot,1 \rangle $ such that
$\mathbf{N}$ is non-trivial: $0\ne 1$
every non-zero element has a multiplicative inverse: $x\ne 0\Longrightarrow \exists y (x\cdot y=1)$
Remark: The inverse of $x$ is unique, and is usually denoted by $x^{-1}$.
Morphisms
Let $\mathbf{M}$ and $\mathbf{N}$ be near-fields. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism:
$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$
Remark: It follows that $h(0)=0$ and $h(-x)=-h(x)$.
Examples
Example 1:
Basic results
$0$ is a zero for $\cdot$: $0\cdot x=0$ and $x\cdot 0=0$.
Properties
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$
