Existence of left invariant measures on a semigroup $S$ with definitions of $m(A)=m(xA)$ or $m({y:y \ \text{belongs to}\ xA})$ does not mean that support $m$ is a right group because it could be embedded in a right group, i.e. the direct product of a semigroup embeddable in a group and right nulls semigroup. But even the later description is not equivalent to existence of a left invariant measure on the left cancellable semigroup because it requires Malcev conditions.