Idea Definition o The ideal generated by a subset H will be denoted by id(H), and if H=af, we write id(a) for id(a) we shall call id(a)a principal ideal. o For an order P, a subset a C P is called down-set fr∈ A and y≤ .c imply that y∈A
Ideal . Definition . . 1. The ideal generated by a subset H will be denoted by id(H), and if H = {a}, we write id(a) for id(a); we shall call id(a) a principal ideal. 2. For an order P, a subset A ⊆ P is called down-set if x ∈ A and y ≤ x imply that y ∈ A. Yi Li (Fudan University) Discrete Mathematics March 5, 2013 6 / 17
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 o I=id(h) if and only if Ⅰ={x|x≤hoU…Uhn-1 for some n≥1and ho,…,hn-1∈H} O For a∈L,id(a)={ cnala∈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 5, 2013 7 / 17