Algebraic Lattices

Let L be a non empty reflexive relational structure. The functor CompactSublatt(L) yields a strict full relational substructure of L and is defined as follows: (Def. 1) For every element x of L holds x ∈ the carrier of CompactSublatt(L) iff x is compact. Let L be a lower-bounded non empty reflexive antisymmetric relational structure. Observe that CompactSublatt(L) is non empty. Next we state three propositions: (1) Let L be a complete lattice and x, y, k be elements of L. If x ¬ k and k ¬ y and k ∈ the carrier of CompactSublatt(L), then x ≪ y. (2) Let L be a complete lattice and x be an element of L. Then ↑x is an open filter of L if and only if x is compact. (3) For every lower-bounded non empty poset L with l.u.b.’s holds CompactSublatt(L) is join-inheriting and ⊥L ∈ the carrier of CompactSublatt(L). Let L be a non empty reflexive relational structure and let x be an element of L. The functor compactbelow(x) yielding a subset of L is defined by: (Def. 2) compactbelow(x) = {y, y ranges over elements of L: x ­ y ∧ y is compact}.