The Lattice of Subsemilattices of a Semilattice

This note makes two observations about lattices of subsemilattices. First, we establish relationship between direct decompositions of such lattices and ordinal sum decompositions of semilattices. Then we give a characterization of the subsemilattice-lattices. Let us recall some terminology. L will always stand for a semilattice, whose operation will be denoted by. a of an arbitrary lattice L is called neutral if m(a; x; y) = M(a; x; y) for all x; y 2 L, where m(a; x; y) = (a ^ x) _ (a ^ y) _ (x ^ y) and M(a; x; y) = (a _ x) ^ (a _ y) ^ (x _ y). Notice that m(a; x; y) M(a; x; y) holds in any lattice. Lemma 1 Let L be a semilattice and L 0 its subsemilattice. Then L 0 is a neutral element of Sub L ii L ? L 0 is a subsemilattice of L and every element of L 0 is comparable with every element of L ? L 0. Proof. Let L 0 be a subsemilattice of L such that L ? L 0 is a subsemilattice of L as well and every element of L 0 is comparable with every element of L ? L 0. We must prove that, for any