Stable theories of Frechet-powers

Elementary theories of Frechet-powers $A^F$ of structures $A$ are investigated. We put a special emphasis on the study of such theories under the condition of stability as well as on constructions of their models containing a given sets $X$ which are minimal in the sense that, the dimensions of independent sets represented in $X$ do not increase. The basis results of the paper are the characterization of forking (Theorem 2) and a theorem on preservation of dimension in $\lambda$-positive envelopes (Theorem 3).