Completely - Irreducible Elements 1 Robert Milewski University of Białystok

(1) For every sup-semilattice L and for all elements x, y of L holds d−eL(↑x∩↑y) = xt y. (2) For every semilattice L and for all elements x, y of L holds ⊔ L(↓x∩↓y) = xu 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 Lop. If X = X ′ and Y = Y ′, then X tY = X ′uY ′. (6) Let L be a poset with g.l.b.’s, X , Y be subsets of L, and X ′, Y ′ be subsets of Lop. If X = X ′ and Y = Y ′, then X uY = X ′tY ′. (7) For every non empty reflexive transitive relational structure L holds Filt(L) = Ids(Lop). (8) For every non empty reflexive transitive relational structure L holds Ids(L) = Filt(Lop).