On purely prime ideals in semilattices

In this paper, we introduce and study the concept called the purely prime ideal for a meet semilattice L which belongs to the class P M IP ∩ P M F P .We obtained various algebraic and topological properties of purely prime ideals of a meet semilattice L. Denote by P(L), the set of all prime ideals, and denote by Spp(L) the set of all purely prime ideals of a meet semilattice L. The set P(L) can be endowed with the well-known topology called spectral topology.In this paper, we extend and study the concept of the spectral topology on the set Spp(L).We study analogue of the conjecture stated in Tarizadeh and Aghajani [24], "every purely prime ideal of a commutative ring is purely maximal ".In fact, for meet semilattices we prove the conjecture is true when L ∈ P M IP ∩ P M F P is bounded a Stone meet semilattice.