Topic: Abstract Algebra
Question: Let Θ be a congruence on a distributive lattice with 0. Define IΘ (subscript Θ) = O/Θ. Prove that IΘ (subscript Θ) is an ideal of D.
Show that for any ideal I of a distributive lattice D with O, that IΘI (subscript ΘI ) = I. Give an example of a congruence Θ on a distributive lattice D with 0 where ΘIΘ (subscript IΘ) ≠ Θ.
Show that if B is a Boolean algebra, then for any congruence Θ of B that ΘIΘ = Θ (subscript IΘ).
Give an example of a complete distributive lattice that is not isomorphic to D(P), the collection of downsets of P, for any poset P and prove.
For a finite poset P prove that P is an order-isomorphism to the poset β(D*(P)) where D*(X) is the collection of downsets of X and β(D)={P| P is a prime ideal, I≠0, I≠D}.