Glossary of Order Theory - D

D

• dcpo. See directed complete partial order.

• A dense poset P is one in which, for all elements x and y in P with x < y, there is an element z in P, such that x < z < y. A subset Q of P is dense in P if for any elements x < y in P, there is an element z in Q such that x < z < y.

• Directed. A non-empty subset X of a poset P is called directed, if, for all elements x and y of X, there is an element z of X such that x ≤ z and y ≤ z. The dual notion is called filtered.

• Directed complete partial order. A poset D is said to be a directed complete poset, or dcpo, if every directed subset of D has a supremum.

• Distributive. A lattice L is called distributive if, for all x, y, and z in L, we find that x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z). This condition is known to be equivalent to its order dual. A meet-semilattice is distributive if for all elements a, b and x, a ∧ b ≤ x implies the existence of elements a' ≥ a and b' ≥ b such that a' ∧ b' = x. See also completely distributive.

• Domain. Domain is a general term for objects like those that are studied in domain theory. If used, it requires further definition.

• Down-set. See lower set.

• Dual. For a poset (P, ≤), the dual order Pd = (P, ≥) is defined by setting x ≥ y if and only if y ≤ x. The dual order of P is sometimes denoted by Pop, and is also called opposite or converse order. Any order theoretic notion induces a dual notion, defined by applying the original statement to the order dual of a given set. This exchanges ≤ and ≥, meets and joins, zero and unit.