Stone Duality for Skew Boolean Algebras with Intersections
- 30 May 2011
With Karin Cvetko Vah.
For the last two months or so I got “distracted” by a topic which is not properly my core interest, namely non-commutative algebra. It was very strange at first, but now that I got used to non-commutative lattices (yes, there is such a thing) it's kind of fun. Anyhow, Karin Cvetko Vah and I worked out Stone duality for skew Boolean algebras with intersections. Classical Stone duality tells us that Boolean algebras are dual to Stone spaces (zero-dimensional compact Hausdorff spaces), and that the generalized Boolean algebras (which are like Boolean algebras without a top element) are dual to Boolean spaces (zero-dimensional locally-compact Hausdorff spaces). Our skew version of duality says that right-handed skew Boolean algebras with intersections are dual to surjective etale maps between Boolean spaces. It is quite a mouthful to say “right-handed skew Boolean algebra with intersections”, let alone get used to it, but in a certain sense this is a very natural non-commutative structure. And we can get rid of the “right-handed” condition to obtain duality for “skew Boolean algebras with intersections”, as well as several other versions. We use the duality to construct a right-handed skew Boolean algebra with intersections which does not have a lattice section. It has been an open question whether such skew lattices exist.
Abstract: We extend Stone duality between generalized Boolean algebras and Boolean spaces, which are the zero-dimensional locally-compact Hausdorff spaces, to a non-commutative setting. We first show that the category of right-handed skew Boolean algebras with intersections is dual to the category of surjective étale maps between Boolean spaces. We then extend the duality to skew Boolean algebras with intersections, and consider several variations in which the morphisms are restricted. Finally, we use the duality to construct a right-handed skew Boolean algebra without a lattice section.
And now I can get back to homotopy type theory and Coq hacking.