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For which classes of meet-continuous domains does the Scott topology coincide with the Lawson topology, and can this coincidence be characterized purely in terms of order-theoretic properties without invoking topological ones?

Related: Lawson duality theorem for continuous lattices, Hofmann-Mislove theorem, Rudin lemma for dcpos

Meet-continuous domains generalize continuous domains and arise naturally in theoretical computer science and logic as semantic models of computation. The Scott topology on such a domain captures the notion of observable computational properties, while the Lawson topology refines it by also encoding a notion of approximation from below and above. The open problem is whether there exists a purely order-theoretic, intrinsic characterization of exactly which meet-continuous domains have these two topologies agreeing, since known sufficient conditions either smuggle in topological assumptions or restrict to narrower subclasses like continuous domains where the answer is classical and well-understood.

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