Every Čech-complete paratopological group is a topological group. That means that for Čech-complete groups you do not have to require the continuity of the inverse, continuity of multiplication suffices. Every manifold is Čech-complete. Using the affirmative answer to Hilbert’s fifth problem we get that every paratopological group on a manifold is actually a Lie group uniquely determined by the topological group structure.

In the spirit of Martin let me give a correct (I hope I have not forgotten anything) definition which is even wronger than the definition without the inverse:

A topological space $G$ with a function $\cdot\colon G^2\to G$ is called an $n$-dimensional *Lie group* if and only if

$G$ is second-countable

There exists an injective, open continuous map $\iota\colon \mathbb{R}^n\to G$

For every $g\in G$ the map $x\mapsto g\cdot x$ is continuous and surjective

There exists $e\in G$ such that $x\mapsto x\cdot e$ is the identity

For every $g\in G$ the map $x\mapsto x\cdot g$ is continuous

$\cdot$ is associative