Abstract: Throughout the paper, K denotes a fixed commutative field. LetF be a field containing Kin its center such that FK is finite dimensional. A finite (partially) ordered set Y together with an order preserving mapping of .Y into the lattice of all subfields of F containing K is called a K-structure for F, thus, for i E Y, there is given a subfield Fi of F and, moreover, KC Fi C Fj for each i k < Z} with Fi = Fj = F,, = F, = G. Given a K-structure .4” for F, the weighted width of Y is defined as the maximum of all possible sums CjEJ dim F,? , where J is a subset of mutually unrelated elements of 9. An Y-space (W, WJ is a right vector space W over F together with an F,-subspace Wi for each ie 9, such that i < j implies Wi C Wj . The weighted dimension of (W, Wi) is the maximum of all dim WF, . For a given K-structure Y, the Y-spaces form an additive category in which the morphisms (W, Wi) --f (II”, W,‘) are F-linear mappings v: W + W satisfying ~JW~ C W,‘, i E 9. Therefore, the concepts of a direct sum and of an indecomposable Y-space are defined. A K-structure 9 is said to be of Jinite type if there is only a finite number of finite dimensional indecomposable Y-spaces. In the case of a classical K-structure, that is Fi = F for all i E Y, L. A. Nazarova and A. V. Roiter [ 151 and M. M. Kleiner [l l] have characterized the structures of finite type. Their results are extended in the following theorem.