TWO PROPERTIES OF EXISTENTIALLY CLOSED COMPANIONS OF STRONGLY MINIMAL STRUCTURES

Аннотация

The proposed article studies some properties of existentially closed companions of strongly minimal
structures. A criterion for the existential closedness of an arbitrary strongly minimal structure is found in the article
and it is proved that the existentially closed companion of any strongly minimal structure is itself strongly minimal.
It also follows from the resulting description that all existentially closed companions of a given strongly minimal
structure form an axiomatizable class whose elementary theory is complete and model-complete and, therefore,
coincides with its inductive and forcing companions.
This is the reason for the importance of the work done and the high international significance of the theorems
obtained in it. Another equally important consequence of this research is the discovery of an important subclass of
strongly minimal theories. It should be noted that a complete description of this class of theories is an independent
and extremely important task.
It is known that natural numbers with the following relation are an example of a strongly minimal structure in
which the existential type of zero is not minimal. Then the method used in the proof of the last theorem shows that
the existentially closed companion of this structure are integers with the following relation.