Exploring prime $ F- $filters of almost distributive lattices
Ali Yahya Hummdi, Noorbhasha Rafi, M. Balaiah, Y. Monikarchana · AIMS Mathematics · 2025
The concept of $ {F}- $filters is introduced in an almost distributive lattice(ADL) and their properties are studied. A set of equivalent conditions is established for every proper $ {F}- $filter of an ADL to become a prime $ {F}- $filter. For any $ {F}- $filter $ {U} $ of an ADL, $ \mathcal{O}^{{F}}({U}) $ is defined, and it is proved that $ \mathcal{O}^{{F}}({U}) $ is an $ {F}- $filter if $ {U} $ is prime. It is also derived that each minimal prime $ {F}- $filter belonging to $ \mathcal{O}^{{F}}({U}) $ is contained in $ {U} $, and $ \mathcal{O}^{{F}}({U}) $ is the intersection of all the minimal prime $ {F}- $filters contained in $ {U}. $ The concept of $ {F}- $normal ADL is defined and characterized in terms of the prime $ {F}- $filters and minimal prime $ {F}- $filters, as well as relative annihilators with respect to $ {F} $.