Abstract:

We study linearly ordered structures and their complete theories. The
main technical tools used in the analysis are condensations, i.e. partitioning the
ordering into convex parts and then studying the quotient structure and that of the
parts. We introduce a uniformly definable condensation relation cδ that decomposes
the ordering into largest convex pieces whose first order theory is simple: they are
either dense or discrete orderings. We study cδ quotient structures that are expansions
of certain simple countable discrete orderings and give a precise description of
those having Cantor Bendixson rank 1. We also use the condensation cδ to prove
that any linear ordering expanded by finitely many unary predicates and equivalence
relations with convex classes is interpretable in a pure linear ordering.
We introduce notions of linear and strong linear binarity for linearly ordered
structures and their complete theories. In the case of a theory, the defining condition
expresses a property of the automorphism group of its saturated model. We prove
that any complete theory of a linear ordering with unary predicates and equivalence
relations with convex classes is strongly linearly binary. The main result states that a
strongly linearly binary structure is definitionally equivalent to a linear ordering with
unary predicates and equivalence relation with convex classes added. In the proof
we give a description of definable sets in any linear ordering with unary predicates
and equivalence relations with convex classes. 