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Home / Lattices and Boolean Algebras / Ideals and lattice | 2nd Lecture

Ideals and lattice | 2nd Lecture

Dual lattice: A non empty subset I of a lattice L is called dual ( or filter) of L if

(i) ∀ a , b ∊ I ⟹ a ˄ b ∊ I

(ii) ∀ a ∊ I , ∀ l ∊ L ⟹ a ˅ l ∊ I

Dual ideal generated by a subset H of L is denoted by [H]

Principle ideal: Let L be a lattice and a ∊ L be any element. The set (a] = {x ∊ L | x ≤ a} forms an ideal of L, is called principle ideal. It is generated by a.

Principle dual ideal: Let L be a lattice and a ∊ L be any element the set [a) = {x ∊ L | a ≤ x} forms a dual ideal of L is called the principle dual ideal. It is generated by a.

Prime ideals: An ideal A of a lattice L is called a prime ideal of L if A is properly contained in L and wherever a ˄ b ∊ A then a ∊ A or b ∊ A.

Dual prime ideal: A proper dual ideal I of a lattice is called a prime ideal if a ˅ b ∊ I ⟹ a ∊ I or b∊ I.

Ideal lattice: The set of all ideals of a lattice L is called ideal lattice of L. It is denoted by I(L).

Theorem: A dual ideal is a convex sublattice.

Or, Let I and D be ideal and dual ideal of a lattice L. Then I ∩ D is a convex sublattice.

Proof: Let I and D be ideal and dual ideal of lattice L. So I and D are sublattice. Hence I ∩ D is a sublattice.

Now let a, b ∊ I ∩ D, where a ≤ b

Then a, b ∊ I and a, b ∊ D

⟹ [a, b] ⊆ I and [a, b] ⊆ D as I and D is convex sublattices.

Thus [a, b] ⊆ I ∩ D or that I ∩ D is convex sublattice of L.

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