Almost Existentially Closed Models in Positive Logic

Abstract

This paper explores the concept of almost positively closed models in the framework of positive logic. To accomplish this, we initially define various forms of the positive amalgamation property, such as h-amalgamation and symmetric and asymmetric amalgamation properties. Subsequently, we introduce certain structures that enjoy these properties. Following this, we introduce the concepts of $\Delta$-almost positively closed and $\Delta$-weekly almost positively closed. The classes of these structures contain and exhibit properties that closely resemble those of positive existentially closed models. In order to investigate the relationship between positive almost closed and positive strong amalgamation properties, we first introduce the sets of positive algebraic formulas $E_T$ and ${\text{Alg}}_T$ and the properties of positive strong amalgamation. We then show that if a model $A$ of a theory $T$ is a $E_{T^{+}\left(A\right)}$-weekly almost positively closed, then $A$ is a positive strong amalgamation basis of $T$, and if $A$ is a positive strong amalgamation basis of $T$, then $A$ is $A{l}_{T^{+}\left(A\right)}$-weekly almost positively closed.