Omega-categorical weakly o-minimal expansions of Boolean lattices

We study omega-categorical weakly o-minimal expansions of Boolean lattices. We show that a structure (A, <, I) expanding a Boolean lattice (A, <) by a finite sequence J of ideals of A closed under the usual Heyting algebra operations is weakly o-minimal if and only if it is omega-categorical, and hence if and only if A/I has only finitely many atoms for every I in J. We propose other related examples of weakly o-minimal omega-categorical models in this framework, and we examine the internal structure of these models.