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.
* Assuming the axiom of choice, a poset is chain complete if and only if it is a dcpo.

Completeness in terms of universal algebra

As explained above, the presence of certain completeness conditions

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