Quasifield

Definition

Let $Q$ be a set with two binary operations $+$ and $\cdot$, and let $Q^* = Q \setminus \set{0}$. The triple $(Q, +, \cdot)$ is a (left) quasifield if the following hold.

1. $(Q, +)$ is a group with identity element $0$.
2. $a \cdot 0 = 0 \cdot a = 0$ for all $a \in Q$.
3. $(Q^*, \cdot)$ is a loop: there is $1 \in Q^*$ with $1 \cdot a = a \cdot 1 = a$ for all $a \in Q^*$, and for $a, b \in Q^*$ the equations $a \cdot x = b$ and $y \cdot a = b$ have unique solutions in $Q^*$.
4. $a \cdot (b + c) = a \cdot b + a \cdot c$ for all $a, b, c \in Q$.
5. For all $a, b, c \in Q$ with $a \ne b$, the equation $a \cdot x = b \cdot x + c$ has exactly one solution $x \in Q$.

A right quasifield is defined in the same way with $(b + c) \cdot a = b \cdot a + c \cdot a$ in clause 4 and $x \cdot a = x \cdot b + c$ in clause 5.