Vaught's conjecture for theories admitting finite monomorphic decompositions
Volume 256 / 2022
Abstract
An infinite linear order with finitely many unary relations (colors), $\langle X,{ \lt }, U_0,\dots ,U_{n-1} \rangle $, is a good colored linear order iff the largest convex partition of the set $X$ refining the partition generated by the sets $U_j$, $j \lt n$, is finite. The class of relational structures which are definable in such structures by formulas without quantifiers coincides with the class of relational structures admitting finite monomorphic decompositions (briefly, FMD structures) introduced and investigated by Pouzet and Thiéry. We show that a complete theory $\mathcal T $ of a relational language $L$ having infinite models has an FMD model iff all models of $\mathcal T $ are FMD, and call such theories FMD theories. For an FMD theory $\mathcal T $ we detect a definable partition of its models, adjoin a family of monomorphic relations to $\mathcal T $ and confirm Vaught’s conjecture, showing that $\mathcal T $ has either one or continuum many non-isomorphic countable models.
