Gelfand duality for compact pospaces

It is well known that the category of compact Hausdorff spaces is dually equivalent to the category of commutative C^\star-algebras. More generally, this duality can be seen as a part of a square of dualities between compact Hausdorff spaces, C^\star-algebras, compact regular frames and de Vries algebras.

Three of these dualities have been extended to dualities between compact pospaces, stably compact frames and proximity frames, the fourth part of the square being lacking.

We propose the category of bounded Archimedean l-semi-algebra to complete what will be the outside square of dualities and to extend to category of C^\star-algebras.