NEIGHBORHOODS IN WEAKLY ORDERED MINIMAL THEORIES

This paper studies neighborhoods, weak orthogonality, and almost orthogonality of complete non-algebraic 1-types in weakly ordered minimal (weakly o-minimal) theories. A neighborhood is introduced as a tool to describe the local properties of type realizations and to generalize the notion of algebraic closure within a type. Their use allows us to distinguish between types and to refine the structure of their interaction. We formulate and prove the main properties of neighborhoods. In particular, it was established that . On the basis of these results, we investigate the relationships between weak and almost orthogonality of types. In particular, we obtain criteria describing their equivalence, symmetry, and behavior for various classes of types (irrational, quasisolitary, and quasirational). Thus, the paper contributes to clarifying and developing the concepts of orthogonality in weakly o-minimal theories. It is also shown that for certain classes of weakly o-minimal theories, weak and almost orthogonality coincide. The results obtained provide new tools for analyzing the geometry of types in weakly o-minimal theories and open perspectives for further research on structures of weakly o-minimal type. In addition, the proposed approaches can be used for comparison with more general classes of theories.