Distributivity (order Theory) - Distributivity For Semilattices

Distributivity For Semilattices

Semilattices are partially ordered sets with only one of the two lattice operations, so that we speak of meet-semilattices or join-semilattices. Given that there is only one binary operation, distributivity obviously cannot be defined in the standard way. Nevertheless, because of the interaction of the single operation with the given order, the following definition of distributivity remains possible. A meet-semilattice is distributive, if for all a, b, and x:

If a ∧ b ≤ x then there exist a' and b' such that a ≤ a', b ≤ b' and x = a' ∧ b' .

This definition is justified by the fact that given any lattice L, the following statements are all equivalent:

• L is distributive as a meet-semilattice
• L is distributive as a join-semilattice
• L is a distributive lattice.

Thus any distributive meet-semilattice in which binary joins exist is a distributive lattice. Distributive join-semilattices are defined dually: a join-semilattice is distributive, if for all a, b, and x:

If x ≤ a ∨ b then there exist a' and b' such that a' ≤ a, b' ≤ b and x = a' ∨ b' .

A join-semilattice is distributive if and only if the lattice of its ideals (under inclusion) is distributive.

This definition of distributivity allows generalizing some statements about distributive lattices to distributive semilattices.