Recent Preprints

1. Frames and Locales: topology without points, 2nd Edition (with Aleš Pultr)
   December 2025, ~590 pages (Springer Basel, to appear in 2026).
   Preface
   
   
2. Structures: A mathematical reader (with Aleš Pultr)
   July 2025, ~680 pages (submitted for publication).
   Preface
   
   
3. Frobenius identities and geometrical aspects of Joyal-Tierney Theorem (with Aleš Pultr)
   July 2024. pdf file
   Abstract: Open and related maps in the point-free context are studied from a consequently geometric perspective: that is, the opens are concrete well-defined subsets, images of localic maps are set-theoretic images f[U], etc.. We present a short proof of Joyal-Tierney Theorem in this setting, a (geometric) characteristic of localic maps that are just complete, and prove that open localic maps also preserve a natural type of sublocales more general than the open ones. A crucial role is played by Frobenius identities that are briefly discussed also in their general aspects.
   
   
4. Continuity and openness of maps on locales by way of Galois adjunctions (with João Areias)
   Preprint 23-27 of DMUC, August 2023. pdf file
   Abstract: This note reviews some adjoint situations, in the algebraic (pointfree) setting of frames and locales, that describe fundamental properties of mappings such as residuation, continuity and openness.