Hierarchy Elements in an Almost Distributive Lattice

In this paper, we introduce hierarchy elements in an almost distributive lattice with respect to a non-empty set and obtain some of their algebraic properties. We characterize initial segments, ideals, and maximal sets in almost distributive lattices in terms of hierarchy sets and prove that the class of hierarchy sets forms a distributive lattice, which is not an induced sublattice. Also, we characterize hierarchy sets using compatible sets in an almost distributive lattice.