Parabolic Tamari Lattices in Linear Type B

We study parabolic aligned elements associated with the type-B Coxeter group and the so-called linear Coxeter element. These elements were introduced algebraically in (M\"uhle and Williams, 2019) for parabolic quotients of finite Coxeter groups and were characterized by a certain forcing condition on inversions. We focus on the type-B case and give a combinatorial model for these elements in terms of pattern avoidance. Moreover, we describe an equivalence relation on parabolic quotients of the type-B Coxeter group whose equivalence classes are indexed by the aligned elements. We prove that this equivalence relation extends to a congruence relation for the weak order. The resulting quotient lattice is the type-B analogue of the parabolic Tamari lattice introduced for type A in (M\"uhle and Williams, 2019). These lattices have not appeared in the literature before.