On Weakly S-Prime Elements of Lattices

Let £ be a bounded distributive lattice and S a join-subset of £. In this paper, we introduce the concept of S-prime elements (resp. weakly S-prime elements) of £. Let p be an element of £ with S ∧p = 0 (i.e. s∧p = 0 for all s ∈ S). We say that p is an S-prime element (resp. a weakly S-prime element) of £ if there is an element s ∈ S such that for all x, y ∈ £ if p ≤ x ∨ y (resp. p ≤ x ∨ y 6= 1), then p ≤ x ∨ s or p ≤ y ∨ s. We extend the notion of S-prime property in commutative rings to S-prime property in lattices.