semifield
There are different definitions of semifield. We give three such which are not equivalent^{} (http://planetmath.org/Biconditional^{}).
Let $K$ be a set with two binary operations^{} “$+$” and “$\cdot $”.

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Semifield $(K,+,\cdot )$ is a semiring^{} where all nonzero elements have a multiplicative inverse.

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Semifield is the algebraic system $(K,+,\cdot )$, where $(K,+)$ is a group (identity^{} $:=0$), the multiplication^{} “$\cdot $” distributes over the addition “$+$”, $K$ the multiplicative identity $:=1$ and all equations $ax=b$ and $ya=b$ with $a\ne 0$ have solutions $x$, $y$ in $K$.

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Semifield $(K,+,\cdot )$ satisfies all postulates^{} of field except the associativity of the multiplication “$\cdot $”.
Title  semifield 

Canonical name  Semifield 
Date of creation  20130322 15:45:46 
Last modified on  20130322 15:45:46 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  7 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 16Y60 
Classification  msc 12K10 
Related topic  NonAssociativeAlgebra 