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semifield
There are different definitions of semifield. We give three such which are not equivalent.
Let $K$ be a set with two binary operations “$+$” and “$\cdot$”.

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

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

Semifield $(K,\,+,\,\cdot)$ satisfies all postulates of field except the associativity of the multiplication “$\cdot$”.
Related:
NonAssociativeAlgebra
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Reference
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