Mdomain

Mdomain is a theoretical construct used in computer science and mathematics to model systems whose state can be understood as a composition of multiple interacting domains. The concept is a generalization of domain theory, introducing a modular structure that captures per-domain information while allowing cross-domain interaction.

In an Mdomain, a carrier set D is equipped with a partial order ≤ that represents information approximation, a least element ⊥ representing undefined or incomplete information, and a coherence structure that ties together the per-domain components.

Typically, an Mdomain consists of a family of per-domain domains {D_i} indexed by a set M of domain identifiers, together with projection maps p_i: D → D_i and compatible injections. The global order is the least upper bound of compatible elements, enabling fixed-point reasoning. A key feature is the compatibility condition: for any i, j, projections preserve order and suprema; domain operations are monotone and continuous.

An example would be a program analyzer that reasons about memory and control-flow domains separately but aggregates results to a global domain M.

Applications include denotational semantics of modular programming languages, security policy modeling in multi-domain systems, and modular verification.

History: The idea emerged in discussions of combining domain theories with modularity in the early 2000s; since then researchers have proposed several variants. Further reading includes texts on domain theory, denotational semantics, and modular semantics, which discuss how multi-domain structures support compositional reasoning in complex systems.