Table of Contents
Semifields
Abbreviation: Sfld
Definition
A \emph{semifield} is a semiring with identity $\mathbf{S}=\langle S,+,\cdot, 1\rangle $ such that
$\langle S^*,\cdot,1\rangle$ is a group, where $S^*=S-\{0\}$ if $S$ has an absorbtive $0$, and $S=S^*$ otherwise.
Morphisms
Let $\mathbf{S}$ and $\mathbf{T}$ be semifields. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\to T$ that is a homomorphism:
$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$
Examples
Example 1:
Basic results
The only finite semifield that is not a field is the 2-element Boolean semifield: https://arxiv.org/pdf/1709.06923.pdf
Properties
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &2
f(3)= &1
f(4)= &1
f(5)= &1
f(6)= &0
\end{array}$
