Left quasifield

Definition

A left quasifield is a set $Q$ equipped with:

• A (infix) binary operation $+$, called the addition or additive operation.
• A unary operation $-$, called the additive inverse.
• A constant $0$, called zero.
• A (infix) binary operation $*$, called the multiplication.
• A constant $1$ such that $1\neq 0$.

Such that:

• $(Q,+,0,-)$ is a group.
• $(Q\setminus \{0\},*,1,)$ is a Moufang loop with multiplication $*$ and identity element $1$.
• $\!a*(b+c)=(a*b)+(a*c)$ (i.e., we have the left distributive law).
• $\!a*x=(b*x)+c$ has exactly one solution in $x$ for any fixed $a,b,c\in Q$ with $a\neq b$.