Scott Domain - Explanation

Explanation

Scott domains are intended to represent partial algebraic data, ordered by information content. An element is a piece of data that might not be fully defined. The statement means " contains all the information that does".

With this interpretation we can see that the supremum of a subset is the element that contains all the information that any element of contains, but no more. Obviously such a supremum only exists (i.e., makes sense) provided does not contain inconsistent information; hence the domain is directed and bounded complete, but not all suprema necessarily exist. The algebraicity axiom essentially ensures that all elements get all their information from (non-strictly) lower down in the ordering; in particular, the jump from compact or "finite" to non-compact or "infinite" elements does not covertly introduce any extra information that cannot be reached at some finite stage. The bottom element is the supremum of the empty set, i.e. the element containing no information at all; its existence is implied by bounded completeness, since, vacuously, the empty set has an upper bound in any non-empty poset.

On the other hand, the infimum is the element that contains all the information that is shared by all elements of, and no less; if contains inconsistent information, then its elements have no information in common and so its infimum is . In this way all infima exist, but not all infima are necessarily interesting.

This definition in terms of partial data allows an algebra to be defined as the limit of a sequence of increasingly more defined partial algebras — in other words a fixed point of an operator that adds progressively more information to the algebra. For more information, see Domain theory.