Joining properties of automorphisms disjoint with all ergodic systems

We study the class $Er{g}^{\perp }$ of automorphisms which are disjoint with all ergodic systems. We prove that the identities are the only multipliers of $Er{g}^{\perp },$ that is, each automorphism whose every joining with an element of $Er{g}^{\perp }$ yields a system which is again an element of $Er{g}^{\perp }$, must be an identity. Despite this fact, we show that $Er{g}^{\perp }$ is closed by taking Cartesian products. Finally, we prove that there are non-identity elements in $Er{g}^{\perp }$ whose self-joinings always yield elements in $Er{g}^{\perp }$. This shows that there are non-trivial characteristic classes included in $Er{g}^{\perp }$.