Abstract
The paper studies a local analogue of the strict order property inside a cut for linearly ordered structures, arising in the theory of order-stable models. A natural attempt to formulate this property in terms of finite c hains o f s trictly n ested d efinable su bsets in tersected wi th re alizations of a cu t is considered. It is shown that such a formulation is not elementary: it may hold in an elementary extension but fail to be preserved in the original model and therefore does not provide a correct local criterion. To overcome this drawback, the notion of strict inclusion of convex components of definable s ets i s i ntroduced, together with its uniform version for families of formulas with parameters. It is proved that the strict order property inside a cut is equivalent to the existence of the uniform property of inclusion of convex components. The obtained result clarifies t he l ocal n a ture o f t his p r operty, s hows i ts d i fference fr om th e cl as sical lo cal strict order property, and makes it applicable to the study of order-stable theories and their models. Thus, the paper complements the criterion of order-stability.