Domainiin

Domainiin is a term used in theoretical computer science and mathematical logic to describe a class of domain-like structures intended to generalize domain theory by incorporating an information inclusion aspect. The term is not widely standardized and may appear in speculative or introductory discussions rather than formal literature. In this sense, domainiin serves as a conceptual framework for examining how state refinement and information content interact in computation.

Formally, a domainiin can be described as a triple (D, ≤, ⊑), where (D, ≤) is a directed-complete partial order with a least element ⊥, and ⊑ is a preorder on D representing information inclusion. The relation is required to be monotone with respect to ≤ (x ≤ y implies x ⊑ y) and compatible with directed suprema: if x_i ⊑ y_i for all i in a directed index set, then sup_i x_i ⊑ sup_i y_i. Additional axioms may be imposed to enforce continuity of the information operator.

Possible applications include modeling systems where states carry both a refinement structure and an intrinsic information content, such as abstract interpretation frameworks, concurrent semantics, or dataflow analyses. Because domainiin is not standard, concrete instances are usually constructed to illustrate the interaction between information order and domain-theoretic properties rather than to serve as an established theory.

Etymology and status: The name combines 'domain' with a suffix reminiscent of other mathematical coinages to denote related objects; there is no widely accepted etymology. The concept remains largely theoretical or pedagogical rather than a standard reference in the literature. See also domain theory, Scott domains, information order.