Minimal self-joinings for nonsingular transformations

The notion of minimal self-joinings for conservative nonsingular actions is defined as a restriction on the nature of rational self-joinings. The need to consider rational joinings is demonstrated by showing that any two type II 1 actions whose Cartesian product is ergodic have type III λ nonsingular joinings. Lastly, actions of all Krieger types with minimal self-joinings are constructed. Hence these actions are prime and commute only with their powers.