Discrete Mathematics ( II ) Spring 2017 Lecture 2 : Lattice ( II ) Lecturer : Yi
نویسنده
چکیده
We have already learned ideal of a ring. A lattice is an algebraic structure which looks like a ring for both have two operations. Similarly, a lattice also has a special structure named as the same as ideal. Definition 1. 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. Specially, A proper ideal I of L is prime if a, b ∈ L and a ∩ b ∈ I imply that a ∈ I or b ∈ I. Specially, {0} and L itself are two trivial ideals. Let’s consider the following example. Example 1. Given a lattice and two sublattices P and I as shown in Figure 1, where P = {a, 0} and I = {0}.
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