The properties of amalgamation and joint embedding in the meaning of positive Jonsson theories

In this paper, we study special subclasses of theories on the basis of the connection between the amalgamation property and the joint embedding property, as well as between the h-amalgamation property and the joint continuation property. Our results are presented in both the classical first-order logic and positive logic, which exhibit a parallel structure. We establish sufficient conditions under which the amalgamation property implies the joint embedding property, and conversely; the h-amalgamation property implies the joint continuation property, and vice versa. Additionally, we investigate the preservation of these subclass links under extensions of the given theory.