Dimdomain

Dimdomain is a concept in domain theory describing a domain D equipped with a dimension function dim: D → N ∪ {∞}, intended to measure the informational or computational complexity of elements. A dimdomain is a pair (D, dim) where D is a partially ordered set with a least element ⊥ that is directed-complete, and dim satisfies monotonicity, Scott-continuity, and a finiteness condition on compact elements. Specifically, dim is monotone: if x ≤ y then dim(x) ≤ dim(y); Scott-continuous: for every directed subset S ⊆ D, dim(sup S) = sup{dim(s) : s ∈ S}; dim(⊥) = 0; and every compact element has finite dimension. These axioms ensure that dimension behaves well with respect to limits and approximation, aligning with the intuition that more informative or refined elements have higher dimension.

Examples include flat domains in which all non-bottom elements have dim 1, and product domains where dim(x, y) = dim(x) + dim(y). Function spaces can be given a dim via a suitable rule, such as dim(f) = sup{dim(f(x)) : x in the domain}, depending on the modeling goal. Dimdomains are used to formalize notions of information content, dependency, or resource usage in denotational semantics and type systems, particularly in analyses that track how data flows through computations. They also appear in abstract interpretation and in studies of dimension-based complexity measures.

The term is primarily encountered in theoretical discussions and is not universally standardized; variants exist depending on the chosen dimension axioms and the specific domain-theoretic framework. See also domain theory, dimension theory, information flow, and denotational semantics.