A dual equivalence of the category of Lawson compact algebraic L -domains and spectrally continuous functions

Abstract We extend Stone duality to domain theory, establishing a logical representation of recursive domains in denotational semantics. Specifically, we prove a dual equivalence between the category $\textbf{LAD}$ of Lawson compact algebraic $L$-domains with spectrally continuous functions and the category $\textbf{NOD}$ of NOD-lattices with NOD-lattice homomorphisms. The duality is constructed via two inverse correspondences: the compact stable open subsets of any Lawson compact algebraic $L$-domain form an NOD-lattice, and the spectrum $\operatorname{Spec}(L)$ of any NOD-lattice $L$ is a Lawson compact algebraic $L$-domain. Moreover, these correspondences are mutually inverse up to isomorphism, yielding two representation theorems: every NOD-lattice is isomorphic to the lattice of compact stable open subsets of its spectrum, and every Lawson compact algebraic $L$-domain is isomorphic to the spectrum of its lattice of compact stable open subsets. This result deepens the structural link connecting semantic domains with lattices admitting finite disjoint decomposition, thereby offering a new tool for the logical analysis of recursive computational structures.