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 non-zero 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 $a\neq 0$ have solutions $x$ , $y$ in $K$ .
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Semifield $(K,\,+,\,\cdot)$ satisfies all postulates of field except the associativity of the multiplication “ $\cdot$ ”.