Hierarchy Sets in Almost Distributive Lattices and Their Structural Properties (European Journal of Pure and Applied Mathematics, 2025)
This paper studies hierarchy sets in almost distributive lattices, focusing on two key types: prime and maximal hierarchy sets. We show that every maximal hierarchy set is prime, but not vice versa, and use Zorn’s Lemma to prove the existence of prime hierarchy sets extending a given one. We also introduce inverted-hierarchy sets, defined via join operations, and analyze their relation to filters. The results provide structural insights and extend ideal and filter theory within almost distributive lattices.
Keywords: Almost distributive lattices; Ideals; Filters; Hierarchy sets; Prime hierarchy sets; Maximal hierarchy sets; Inverted-hierarchy sets