Vaught's conjecture for monomorphic theories

Abstract

A structure Y of a relational language L is called almost chainable iff there are a finite set F ⊂ Y and a linear order < on the set Y \F such that for each partial automorphism ϕ (i.e., local automorphism, in Fraïssé's terminology) of the linear order Y \ F, < the mapping id F ∪ϕ is a partial automorphism of Y. By a theorem of Fraïssé, if |L| < ω, then Y is almost chainable iff the profile of Y is bounded; namely, iff there is a positive integer m such that Y has ≤ m non-isomorphic substructures of size n, for each positive integer n. A complete first order L-theory T having infinite models is called almost chainable iff all models of T are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of T . In addition, it is proved that an almost chainable theory has either one or continuum many non-isomorphic countable models and, thus, the Vaught conjecture is confirmed for almost chainable theories.

Vaught's conjecture for theories admitting finite monomorphic decompositions

“…By the well-known result of Rosenstein [10], the class C of countable ω-categorical linear order types is the smallest class containing 1 and closed under finite sums and the shuffle operation; in addition (see [11, p. 299]) ( * ) The class C is closed under convex substructures and reverses. Now, if the structure Y is ω-categorical, then, by [6,Theorem 1.2], there is an ω-categorical linear order X = Y, < , where < ∈ L Y ; so, τ := otp(X) ∈ C. By a result of Gibson, Pouzet and Woodrow [3, Theorem 9], which is a generalization of similar results from [2] and [4], we have one of the following three cases.…”

“…Thus g ∈ Pa(X ) and, by our assumption, g ∈ Pa(Y ), that is, g : Y K → Y H is an isomorphism. So, by (7), f = G −1 •g •F : Y K → Y H is an isomorphism and belongs to Pa(Y). By Fact 2.6 we have Pa(X ) ⊂ Pa(Y).…”

“…In particular, some model of T is monomorphic iff all models of T are monomorphic and, more generally, some model of T is almost chainable iff all models of T are almost chainable (see also [6] and [7]).…”

“…If n(T ) = 0, then each model Y of T is monomorphic, Ker(Y) = ∅ and we define d(T ) = 1 (see the end of the previous subsection). Namely, if Y |= T , then, by Theorem 4.3, Y is an FMD structure and we have: T is almost chainable iff (by [7], see also Example 3.5) Y is almost chainable iff (see [9, p. 9, the sentence before Theorem 1.7]) d(Y) = 1 iff (by Theorem 4.13) d(T ) = 1.…”

“…Generally speaking, if Vaught's conjecture is confirmed for the (theories of) structures belonging to a class K, it is natural to ask whether that result can be extended to the theories of the structures which are (first-order) definable (or, more generally, interpretable) in the structures from K. In this sense, the main result of [6] (resp. [7]) extends Theorem 1.1 to the class of theories of relational signatures having a model Y which is definable in some linear order (resp. in some linear order with finitely many "unary singletons at the beginning"), X, by formulas without quantifiers (Π 0 -formulas).…”