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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:ST that is a homomorphism:

h(x+y)=h(x)+h(y), h(xy)=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

Subclasses

Superclasses

References


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