We continue the analysis of foundations of positive model theory as introduced by Ben Yaacov and Poizat. The objects of this analysis are $h$-inductive theories and their models, especially the “positively” existentially closed ones. We analyze topological properties of spaces of types, introduce forms of quantifier elimination, and characterize minimal completions of arbitrary $h$-inductive theories. The main technical tools consist of various forms of amalgamations in special classes of structures.

We study the amalgamation property in positive logic, where we shed light on some connections between the amalgamation property, Robinson theories, model-complete theories and the Hausdorff property.

In this paper we extend of the notion of algebraically closed given in the case of groups and skew fields to an arbitrary h-inductive theory. The main subject of this paper is the study of the notion of positive algebraic closedness and its relationship with the notion of positive closedness and the amalgamation property.

We introduce the notions of weakly and locally positive Robinson theories. We give a characterization of weakly positive Robinson theories by the amalgamation property, and a syntactic characterization of locally positive Robinson theories.