Discrete mathematics Yi Li Software school Fudan universit March 5. 2013
. . Discrete Mathematics Yi Li Software School Fudan University March 5, 2013 Yi Li (Fudan University) Discrete Mathematics March 5, 2013 1 / 17
Review o Review of a partial order set Review of abstract algebra Lattice and sublattice
Review Review of a partial order set Review of abstract algebra Lattice and Sublattice Yi Li (Fudan University) Discrete Mathematics March 5, 2013 2 / 17
utline o Special Lattices o Boolean Algebra
Outline Special Lattices Boolean Algebra Yi Li (Fudan University) Discrete Mathematics March 5, 2013 3 / 17
Idea efinition(Ring Given a ring R and a nonempty set I CR. I is an ideal of R if it subjects to o For any a,b∈I,a-b∈1. o For any a∈I,r∈R,ar,ra∈1 Definition(Lattice) a subset of a lattice is an ideal if it is a sublattice of L and a∈ I and a∈ imply that a∩a∈ A proper ideal I of L is prime if a,b∈ L and a∩b∈I imply that a∈rorb∈I
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 5, 2013 4 / 17
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: Ideal 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 5, 2013 5 / 17