Idea 「The eorem Let l be a lattice and let H and i be nonempty subsets o I is an ideal if and only if the following two conditions hold oa,b∈ I implies that a∪b∈I, o I is a down-set oI=id(h) if and only if I={x|x≤hoU…Uhn-1 for some n>1and he han-1∈H} O For a∈L,id(a)={x∩ax∈L}Ideal Theorem Let L be a lattice and let H and I be nonempty subsets of L. 1 I is an ideal if and only if the following two conditions hold: 1 a, b ∈ I implies that a ∪ b ∈ I, 2 I is a down-set. 2 I = id(H) if and only if I = {x|x ≤ h0 ∪ · · · ∪ hn−1 for some n ≥ 1 and h0, . . . , hn−1 ∈ H}. 3 For a ∈ L, id(a) = {x ∩ a|x ∈ L}. Yi Li (Fudan University) Discrete Mathematics March 6, 2012 7 / 1