Completeness (order Theory) - Relationships Between Completeness Properties

Relationships Between Completeness Properties

It was already observed that binary meets/joins yield all non-empty finite meets/joins. Likewise, many other (combinations) of the above conditions are equivalent.

• The best-known example is the existence of all suprema, which is in fact equivalent to the existence of all infima. Indeed, for any subset X of a poset, one can consider its set of lower bounds B. The supremum of B is then equal to the infimum of X: since each element of X is an upper bound of B, sup B is smaller than all elements of X, i.e. sup B is in B. It is obvious that it is the greatest element of B and hence the infimum of X. In a dual way, the existence of all infima implies the existence of all suprema.

• Bounded completeness can also be characterized differently. By an argument similar to the above, one finds that the supremum of a set with upper bounds is the infimum of the set of upper bounds. Consequently, bounded completeness is equivalent to the existence of all non-empty lower bounded infima.

• A poset is a complete lattice if and only if it is a cpo and a join-semilattice. Indeed, for any subset X, the set of all finite suprema (joins) of X is directed and the supremum of this set (which exists by directed completeness) is equal to the supremum of X. Thus every set has a supremum and by the above observation we have a complete lattice. The other direction of the proof is trivial.

• The following equivalence requires the Axiom of Choice:

A poset is chain-complete if and only if it is a dcpo.