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

We study the set $\text{sncb}(p,q)$ of annular non-crossing permutations of type B, and we introduce a corresponding set $\text{ncb}(p,q)$ of annular non-crossing partitions of type B, where $p$ and $q$ are two positive integers. We prove that the natural bijection between $\text{sncb}(p,q)$ and $\text{ncb}(p,q)$ is a poset isomorphism, where the partial order on $\text{sncb}(p,q)$ is induced from the hyperoctahedral group $B_{p+q}$, while $\text{ncb}(p,q)$ is partially ordered by reverse refinement. In the case when $q=1$, we prove that $\text{ncb}(p,1)$ 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 $\text{sncd}(p,q)$ and $\text{ncd}(p,q)$. For $q=1$, the poset $\text{ncd}(p,1)$ coincides with a poset ``$N{C}^{(D)}(p+1)$‘’ constructed in a paper by Athanasiadis and Reiner in 2004, and is a lattice by the results of that paper.