Posets of annular non-crossing partitions of types B and D

We study the set S"n"c^B(p,q) of annular non-crossing permutations of type B, and we introduce a corresponding set NC^B(p,q) of annular non-crossing partitions of type B, where p and q are two positive integers. We prove that the natural bijection between S"n"c^B(p,q) and NC^B(p,q) is a poset isomorphism, where the partial order on S"n"c^B(p,q) is induced from the hyperoctahedral group B"p"+"q, while NC^B(p,q) is partially ordered by reverse refinement. In the case when q=1, we prove that NC^B(p,1) is a lattice with respect to reverse refinement order. We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between S"n"c^D(p,q) and NC^D(p,q). For q=1, the poset NC^D(p,1) coincides with a poset ''NC^(^D^)(p+1)'' constructed in a paper by Athanasiadis and Reiner [C.A. Athanasiadis, V. Reiner, Noncrossing partitions for the group D"n, SIAM Journal of Discrete Mathematics 18 (2004) 397-417], and is a lattice by the results of that paper.