Definition: An ideal A of a lattice L is called a prime ideal of L if A is properly contained in L and whenever a ^ b ∈ A then a ∈ A or b ∈ B. Theorem: Prove that a lattice L is a chain if and only if ...

Read More »## Complements| Lattice

Complements: Let x, y be any elements of a lattice L. If x ˄ y =0 and x ˅ y = 1 then we say y is a complements of x. Or, Let [a, b] be an interval in a lattice L. Let x ∊ [a, b] be any element. ...

Read More »## The set of all ideals of a lattice L forms a lattice under ⊆ relation.

Or, Suppose L is a lattice, What do you mean by I(L), prove that (I(L), ⊆) is a lattice. Proof: Let I(L) is a set of all ideals of a lattice L. Then clearly, L ∊ I(L) ∴ I(L) is non empty. Now, let us first, show that (I(L), ⊆) ...

Read More »## 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 ...

Read More »## Ideals | Lattices and Boolean algebra

Definition: A non empty subset I of a lattice L is called an ideal of L if (i) a, b ∊I ⟹a ˅ b ∊ I (ii) a ∊ I, l ∊ L ⟹ a ˄ l ∊ I. Example: Let {1, 2, 5, 10} be a lattice of factors ...

Read More »## Sublattice and Convex sublattice

Sublattice: Let (L, ˄, ˅) be a lattice. A non empty subset S of L is called a Sublattice of L if S itself is a lattice under same operations ˄ and ˅ in L. Or, A non empty subset S of a lattice L is called a Sublattice of ...

Read More »## Complete lattices

Definition: A lattice L is called a complete lattice if every non empty sub set of L has its Sup and Inf in L. Theorem: Dual of a complete lattice is complete. Proof: Let (L, ?) be a complete lattice and let (L, ?) be its dual. Then (L, ?) ...

Read More »## Define two definitions of lattices and show that two definitions of lattices is equivalence.

Definition I(according to poset): A poset (L, ≤) is said to form a lattice if for every a, b ∈ L, Sup{a, b} and Inf{a, b} exist in L. In that case, we write Sup{a, b} = a ˅ b Inf{a, b} = a ˄ b Definition II(algebraic): A non ...

Read More »## Show that the cardinal product of two self dual posets is self dual

Proof: Let A and B be the given self dual posets. Let f: A→A and g: B → B be the isomorphisms. Now we define h: A × B → A × B, such that, h((a, b)) = (f(a), g(b)) then h is well defined, one – one map as ...

Read More »## Show that product of two posets is a poset

Proof: Let A and B be two posets . We have to show that A × B = {(a, b) | a ∈ A, b ∈ B} Forms a poset under the relation defined by (a1, b1) ≤ (a2, b2) ⇔ a1 ≤ a2 in A and b1 ≤ b2 ...

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