Left-algebraic group

Definition

A left-algebraic group over a field $K$ is a set $G$ equipped with the following two structures:

• The structure of an algebraic variety over the field $K$.
• The structure of a group

such that the following compatibility conditions are satisfied:

Operation	Arity	Condition	Comments
Multiplication or product	2 (so it's a map $G\times G\to G$)	For any fixed $g\in G$, the map $x\mapsto gx$ is a regular morphism from $G$ to itself (i.e., it is algebraic).
Identity element	0 (it's a constant element $e\in G$)	no condition. As such, we may impose the condition that the map from a one-point variety to $G$ sending the point to the identity element is a regular morphism, but this condition is vacuously true.
Inverse map	1 (so it's a map $G\to G$)	no condition.