## Semantic spaces in Priestley form

### Mohamed El-Zawawy

Institute of Cybernetics

Thursday, 4 December 2008, 14:00

Cybernetica Bldg (Akadeemia tee 21), room B101

Slides from the talk [pdf]

**Abstract**: In 1937 Marshall Stone extended his celebrated
representation theorem for Boolean algebras to distributive
lattices. In modern terminology, the representing topological spaces
are zero-dimensional stably compact, but typically not Hausdorff. In
1970, Hilary Priestley realised that Stone's topology could be
enriched to yield orderdisconnected compact ordered spaces.

In the this talk, I generalise Priestley duality to a
representation theorem for strong proximity lattices. For these a
"Stone-type" duality was given in 1995 in joint work between Philipp
Sünderhauf and Achim Jung, which established a close link between
these algebraic structures and the class of all stably compact
spaces. The feature which distinguishes the present work from this
duality is that the proximity relation of strong proximity lattices is
"preserved" in the dual, where it manifests itself as a form of
"apartness." This suggests a link with constructive mathematics which
in this talk we can only hint at. Apartness seems particularly
attractive in view of potential applications of the theory in areas of
semantics where continuous phenomena play a role; there, it is the
distinctness between different states which is observable, not
equality.

The idea of separating states is also taken up in our discussion of
possible morphisms for which the representation theorem extends to an
equivalence of categories.

(Joint work with Achim Jung.)

Tarmo Uustalu

Last update 4.12.2008