Exploring prime $F-$filters of almost distributive lattices

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 $F$.