Discrete mathematics Yi Li Software school Fudan universit March 6. 2012
Discrete Mathematics Yi Li Software School Fudan University March 6, 2012 Yi Li (Fudan University) Discrete Mathematics March 6, 2012 1 / 1
Review o Review of a partial order set o Review of abstract algebra o Lattice and Sublattice
Review Review of a partial order set Review of abstract algebra Lattice and Sublattice Yi Li (Fudan University) Discrete Mathematics March 6, 2012 2 / 1
utline Special Lattices o Boolean Algebra
Outline Special Lattices Boolean Algebra Yi Li (Fudan University) Discrete Mathematics March 6, 2012 3 / 1
Idea Definition(Ri Given a ring R and a nonempty set I CR. I is an ideal of R if it subjects to For any a,b∈I,a-b∈1. For any a∈I,r∈R,ar,Ta∈I. Definition( Lattice) A subset of a lattice is an ideal if it is a sublattice of L and x∈ I and a∈ L imply that a na∈I A proper ideal I of L is prime if a,b∈ L and a∩b∈I imply that a∈Iorb∈r
Ideal Definition (Ring) Given a ring R and a nonempty set I ⊆ R. I is an ideal of R if it subjects to: 1 For any a, b ∈ I, a − b ∈ I. 2 For any a ∈ I, r ∈ R, ar, ra ∈ I. Definition (Lattice) A subset I of a lattice L is an ideal if it is a sublattice of L and x ∈ I and a ∈ L imply that x ∩ a ∈ I. A proper ideal I of L is prime if a, b ∈ L and a ∩ b ∈ I imply that a ∈ I or b ∈ I. Yi Li (Fudan University) Discrete Mathematics March 6, 2012 4 / 1
Idea am dle Given a lattice and sublattice p and i as shown in the following Figure, where P=a,0 and I=0f Figure: ldeal and prime ideal
Ideal Example Given a lattice and sublattice P and I as shown in the following Figure, where P = {a, 0} and I = {0}. 0 a b 1 I P Figure: Ideal and prime ideal Yi Li (Fudan University) Discrete Mathematics March 6, 2012 5 / 1