On the n-uniqueness of types in rosy theories

We prove that in a rosy theory, the n-uniqueness of a strong type p implies the strong n-uniqueness of p. In addition we study some of the boundary properties of p. In [5], given a strong type p = tp(c/A) in a stable theory T , if T has (≤ n)-uniqueness while p does not have (n + 1)-uniqueness then, over an infinite Morley sequence of the type of some finite tuple c′ ∈ acl(cA) over A witnessing the failure of (n+1)-uniqueness of p, a canonical way to construct an n-ary polygroupoid is suggested (*). This construction method crucially used in establishing the Hurewicz correspondence in [6]. Now a question remains as to whether the assumption above that T has (≤ n)-uniqueness can simply be reduced to that p has (≤ n)uniqueness. It is straightforward to see that in order to draw (*) what is used in [5] is indeed only the assumption that p has (≤ n)-uniqueness for each k ≥ 1 where p is the complete type of k independent realizations of p over A. Now in this paper we prove that p has n-uniqueness iff p has n-uniqueness for each k ≥ 1, answering the question positively. We work with a fixed rosy theory T (unless said otherwise). If T is simple then we work in M = M (so tuples and sets are hyperimaginaries) of T and independence is nonforking independence. But if T is non-simple rosy then we work in M = M and independence refers to thorn-nonforking. In particular, cl(A) is bdd(A) in simple T , and is acl(A) in non-simple rosy T . Of course if T is stable or supersimple then the pairs of corresponding notions all coincide [2]. For the theoretical background of rosy, simple or stable theories, we refer the reader to [2] or [7]. We also fix a complete type (of possibly an infinite arity) p over A = cl(A). We let CA denote the category of all closed small subsets of M containing A, where morphisms are A-elementary maps (i.e., fixing A pointwise). This work was supported by NRF of Korea grant 2013R1A1A2073702. 1