  A paper by Cavenagh and Wanless diagnosed the possible intersection of any
two transversals of the back circulant Latin square B_n, and used the result to
completely determine the spectrum for 2-way k-homogeneous latin trades. We give
a generalization of this problem for the intersection of \mu transversals of
B_n and provide a construction for this problem, as well as providing base
designs for the construction in the cases \mu= 3, 4 found by a computational
search. This result is then applied to the problem of finding \mu-way
k-homogeneous Latin trades.
  We generalize this problem to the intersection of \mu transversals of B_n
such that the transversals intersect stably (that is, the intersection of any
pair of transversals is independent of the choice of the pair) and show that
these structures can be used to construct \mu-way k-homogeneous circulant latin
trades of odd order. We provide a number of basic existence and non-existence
results for \mu transversals of B_n that intersect stably, as well as the
results of a computational search for small n. This is followed by the
principal results of this paper; a construction that covers a large portion of
the spectrum when n is sufficiently large, which requires certain base designs.
These base designs are provided in the cases \mu= 3,4, which were found by a
computational search. We use this result to find the existence of \mu-way
k-homogeneous circulant latin trades of odd order, for \mu= 3, 4.
