−Table of Contents
Semifields
Abbreviation: Sfld
Definition
A \emph{semifield} is a semiring with identity S=⟨S,+,⋅,1⟩ such that
⟨S∗,⋅,1⟩ is a group, where S∗=S−{0} if S has an absorbtive 0, and S=S∗ otherwise.
Morphisms
Let S and T be semifields. A morphism from S to T is a function h:S→T that is a homomorphism:
h(x+y)=h(x)+h(y), h(x⋅y)=h(x)⋅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
f(1)=1f(2)=2f(3)=1f(4)=1f(5)=1f(6)=0