EXAMPLES OF LINEAR ORDERS WITH A DEFINABLE UNARY FUNCTION AND THE INDEPENDENCE PROPERTY

After the appearance of the concept of o-minimality, which was introduced by L. van den Dries for expansions of the ordered field of real numbers and generalized to arbitrary linear orders by A. Pillay and C. Steinhorn, linearly ordered structures became firmly established in the circle of interests of specialists in model theory. Numerous generalizations of the concept of o-minimality have appeared in the works of various authors, such as weak o-minimality, quasi-o- minimality, weak quasi-o-minimality, dp-minimality, and o-stability. B. S. Baizhanov and V. V. Verbovskiy proved that o-stability generalizes all the above concepts for linearly ordered structures and that o-stability entails the absence of the independence property. They also proved that any linear order has an o-superstable theory. V. V. Verbovskiy studied o-stable ordered groups, in particular, he proved that they are commutative. In this paper, we begin the study of the question of how complex the theory of a linear order with one unary function can be. We construct an example of an expansion of a linearly ordered structure with one unary function, which has the independence property.