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:

$h(x+y)=h(x)+h(y)$, $h(x\cdot y)=h(x)\cdot h(y)$

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

Classtype No Unbounded No

Finite members

$\begin{array}{lr} f(1)= &1 f(2)= &2 f(3)= &1 f(4)= &1 f(5)= &1 f(6)= &0 \end{array}$