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Near-fields
Abbreviation: NFld
Definition
A \emph{near-field} is a near-rings with identity N=⟨N,+,−,0,⋅,1⟩ such that
N is non-trivial: 0≠1
every non-zero element has a multiplicative inverse: x≠0⟹∃y(x⋅y=1)
Remark: The inverse of x is unique, and is usually denoted by x−1.
Morphisms
Let M and N be near-fields. A morphism from M to N is a function h:M→N that is a homomorphism:
h(x+y)=h(x)+h(y), h(x⋅y)=h(x)⋅h(y)
Remark: It follows that h(0)=0 and h(−x)=−h(x).
Examples
Example 1:
Basic results
0 is a zero for ⋅: 0⋅x=0 and x⋅0=0.
Properties
Finite members
f(1)=1f(2)=f(3)=f(4)=f(5)=f(6)=