O-ideals in Almost Distributive Lattices

Concept of O-ideals is introduced in an Almost Distributive Lattice and these O-ideals are characterized in terms of minimal prime ideals. A necessary and sufficient condition for the zero ideal of an ADL to become an O-ideal is derived. $\star$-ADLs are characterized in terms of their O-ideals and $α$-ideals. Finally, a set of equivalent conditions are established for every O-ideal of an ADL to become a principal ideal.