ORDERED STABILITY AND EXPANSIONS OF A PURE LINEAR ORDER BY A UNARY FUNCTION

A linearly ordered structure is said to be o-stable if each of its Dedekind cut has a “small” number of extensions to complete 1-types. This concept, which was introduced by B.S. Baizhanov and V.V. Verbovsky, generalizes such widely known concepts among specialists in model theory as weak o-minimality, (weak) quasi-o-minimality and dp-minimality of ordered structures. It is based on a combination of the concepts of o-minimality and stability. As we know, the elementary theory of any pure linear order is o-superstable. Indeed, this follows from the fact, which Rubin proved in the late 70s of the 20th century, that any type of one variable is determined by its cut and definable subsets, distinguished by unary predicates or formulas with one free variable. In this paper, we explore the question of what happens if a pure linear order is expanded with a unary function. Two examples were constructed when o-stability is violated; in addition, sufficient conditions for preserving o-stability with such language expansion were found. Research work on this topic is not yet finished, ideally, it would be good to find a criterion for preserving ordered stability when enriching a structure with pure linear order with a new function of one variable.