The category of join-semilattices is a monoidal category with a "classical" tensor product $\otimes$ (analogous to other binary commutative algebras, e.g. abelian groups). JSL is dual to the category Stone(JSL) of continuous join-semilattices that are Stone spaces, and Stone(JSL) also has a tensor product extending continuous bilinear maps to continuous homomorphisms.
For the tensor product of join-semilattices $D, D''$ which are distributive lattices it is well-known that the dual Priestley space of the tensor product $D \otimes D'$ is the given by the product $\hat D \times \hat{D'}$ of the dual Priestley spaces.
Now, I have trouble to come up with a satisfying description of the dual monoidal structure $\hat \otimes$ of the tensor product $\otimes$ of for general join-semilattices. So,
(1) is $\hat \otimes$ isomorphic to the tensor product of Stone join-semilattices, and if not
(2) is there Stone(JSL)-internal description of $\hat \otimes$
Cheers
Literature:
- A standard reference on this duality is the book by Hofmann, Mislove and Stralka [https://doi.org/10.1007/BFb0065929]
- On the tensor product of semilattices there are papers by Fraser: [https://doi.org/10.1007/BF02485362] goes in this direction by giving in the first Lemma a description of finitely generated Bi-ideals (which are elements of the dual of the tensor product), and another paper considers the easy case of the tensor product of the underlying join-semilattices of distributive lattices