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A continuous lattice may be defined as a complete lattice in which arbitrary meets distribute over directed joins. A continuous lattice is naturally regarded as an algebraic structure where the operations are arbitrary meets and directed joins, or equivalently finite meets and limits (in the Lawson topology). Lawson's lemma states that every continuous lattice embeds (preserving these operations) into a power of the unit interval [0, 1].

Is there some type of structure on [0, 1] commuting with the continuous lattice operations, yielding a Stone-type duality for continuous lattices?

More precisely, I am looking for some type of structure (call it "widgets") on [0, 1] such that:

  1. the continuous lattice operations on [0, 1] are widget homomorphisms;
  2. for every continuous lattice $L$, the natural evaluation homomorphism $L \to \mathbf{Widget}(\mathbf{ContLat}(L, [0, 1]), [0, 1])$ is an isomorphism (Lawson's lemma says that it is injective);
  3. for every widget $W$, the natural evaluation homomorphism $W \to \mathbf{ContLat}(\mathbf{Widget}(W, [0, 1]), [0, 1])$ is an isomorphism.

The following algebraic operations on [0, 1] commute with the continuous lattice structure: finite meets, and unary continuous monotone functions which fix 1. Are these enough?

(I suspect some type of metric completeness is also needed, but I don't have a counterexample.)

Note: there are dualities based on the poset $2 = \{0 < 1\}$ that result from forgetting some of the structure of continuous lattices. For example, if we forget about directed meets, then continuous lattices are dual to stably continuous semilattices (Johnstone, Stone spaces, VII 2.12). However, I'm really only interested in a duality that remembers the topological structure of continuous lattices (so perhaps it would be more appropriate to say "Gelfand-type" instead of "Stone-type").

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  • $\begingroup$ I think continuous lattices are dual to Lawson semilattices which are certain compact semilattices with small semilattices. Look at the compendium of continuous lattice book. $\endgroup$ – Benjamin Steinberg Sep 3 '16 at 15:34
  • $\begingroup$ @Benjamin, I think that's a characterization of continuous lattices, not something dual to them. I couldn't find "Lawson semilattices" in the compendium, but according to Stone spaces they are continuous lattices equipped with the Lawson topology. $\endgroup$ – Ronnie Sep 3 '16 at 21:59
  • $\begingroup$ Maybe. It has been a while. $\endgroup$ – Benjamin Steinberg Sep 4 '16 at 0:02

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