Homeomorphism groups on the positive real numbers defined by binary operations

Given a continuous binary operation * on the positive real numbers R+ with ordinary topology, we consider the homeomorphism group Homeo(*)(R+) consisting of all homeomorphisms on R+ which preserve *. We show that if * is any continuous cancellative semigroup operation on R+, then Homeo(*)(R+) is isomorphic to either one of the three well-known groups. Furthermore, we reveal homeomorphism groups defined by some continuous noncancellative semigroup operations on R+.