Completely-Irreducible Elements

The following propositions are true: (1) For every sup-semilattice L and for all elements x, y of L holds ⌈⌉L(↑x∩ ↑y) = x ⊔ y. (2) For every semilattice L and for all elements x, y of L holds ⊔ L(↓x∩↓y) = x ⊓ y. (3) Let L be a non empty relational structure and x, y be elements of L. If x is maximal in (the carrier of L) \ ↑y, then ↑x \ {x} = ↑x ∩ ↑y. (4) Let L be a non empty relational structure and x, y be elements of L. If x is minimal in (the carrier of L) \ ↓y, then ↓x \ {x} = ↓x ∩ ↓y. (5) Let L be a poset with l.u.b.’s, X, Y be subsets of L, and X ′, Y ′ be subsets of L. If X = X ′ and Y = Y ′, then X ⊔ Y = X ′ ⊓ Y ′. (6) Let L be a poset with g.l.b.’s, X, Y be subsets of L, and X ′, Y ′ be subsets of L. If X = X ′ and Y = Y ′, then X ⊓ Y = X ′ ⊔ Y ′. (7) For every non empty reflexive transitive relational structure L holds Filt(L) = Ids(L).