Vaught’s conjecture for theories admitting finite monomorphic decompositions

Abstract

. An infinite linear order with finitely many unary relations (colors), (cid:104) X, <, U 0 , . . . , U n − 1 (cid:105) , 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 < n , is finite. The class of relational structures which are definable in such structures by formulas without quantifiers coin-cides 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 T of a relational language L having infinite models has an FMD model iff all models of T are FMD, and call such theories FMD theories. For an FMD theory T we detect a definable partition of its models, adjoin a family of monomorphic relations to T and confirm Vaught’s conjecture, showing that T has either one or continuum many non-isomorphic countable models.