# 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 $1 \ne 0$.

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 $a \ne b$.