Compactness in positive logic

With the aim of developing the concepts of positive logic and in response to a question that was asked by Poizat in one of his articles, I wrote this article. The main topic is the study of compactness in the extension as a compact structure. they are based on a fundamental characterization of the compactness, which was given by Itai Ben Yacov and Poizat in section Basis of positive logic.