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

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.