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