Examples
- Every finite poset is directed complete and algebraic. Thus any bounded complete finite poset trivially is a Scott domain.
- The natural numbers with an additional top element ω constitute an algebraic lattice, hence a Scott domain. For more examples in this direction, see the article on algebraic lattices.
- Consider the set of all finite and infinite words over the alphabet {0,1}, ordered by the prefix order on words. Thus, a word w is smaller than some word v if w is a prefix of v, i.e. if there is some (finite or infinite) word v' such that w v' = v. For example 10 ≤ 10110. The empty word is the bottom element of this ordering and every directed set (which is always a chain) is easily seen to have a supremum. Likewise, one immediately verifies bounded completeness. However, the resulting poset is certainly missing a top having many maximal elements instead (like 111... or 000...). It is also algebraic, since every finite word happens to be compact and we certainly can approximate infinite words by chains of finite ones. Thus this is a Scott domain which is not an algebraic lattice.
- For a negative example, consider the real numbers in the unit interval, ordered by their natural order. This bounded complete cpo is not algebraic. In fact its only compact element is 0.
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