On ◁∗-maximality Original Research Article

This paper investigates a connection between the semantic notion provided by the ordering ◁∗ among theories in model theory and the syntactic (N)SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1. It is shown here that SOP3 implies SOP2 implies SOP1. In Shelahʹs article (Ann. Pure Appl. Logic 80 (1996) 229) it was shown that SOP3 implies ◁∗-maximality and we prove here that ◁∗-maximality in a model of GCH implies a property called SOP2″. It has been subsequently shown by Shelah and Usvyatsov that SOP2″ and SOP2 are equivalent, so obtaining an implication between ◁∗-maximality and SOP2. It is not known if SOP2 and SOP3 are equivalent. Together with the known results about the connection between the (N)SOPn hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classification of unstable theories without the strict order property.