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Making the ring of continuous localic real functions into a subring of all localic real functions
by
Javier Gutiérrez García
Departamento de Matemáticas, Universidad del País Vasco-Euskal Herriko Unibersitatea, Apdo. 644, Bilbao, Spain
Coauthors: Tomasz Kubiak and Jorge Picado
So far, in pointfree topology, the lattice-ordered ring of all continuous real functions on a frame L has not been a part of the lattice of all lower (or upper) semicontinuous real functions on L [5]. Our goal is to demonstrate a framework in which all those (semi)continuous functions arise (up to isomorphism) as members of the lattice-ordered ring of all frame homomorphisms from the frame of reals into the frame of all congruences on L. That ring is a poinfree counterpart of the ring of all real-valued functions on a topological space, thereby providing a pointfree analogue of the concept of an arbitrary (not necessarily (semi)continuous) real function on L. One feature of this remarkable conception is that one eventually has: lower semicontinuous + upper semicontinuous = continuous. We document its importance by showing how nicely can the insertion, extension and regularization theorems of [1, 2, 3] be recast in the new setting.
This talk is a presentation of much of the paper [4].
[2] J. Gutiérrez García, T. Kubiak and J. Picado, Lower and upper regularizations of frame semicontinuous real functions, Algebra Universalis, to appear.
[3] J. Gutiérrez García, T. Kubiak and J. Picado, Pointfree forms of Dowker's and Michael's insertion theorems, J. Pure Appl. Algebra, to appear.
[4] J. Gutiérrez García, T. Kubiak and J. Picado, Localic real functions: a general setting, Preprint 08-20 of DMUC, April 2008 (submitted for publication).
[5] J. Gutiérrez García and J. Picado, On the algebraic representation of semicontinuity, J. Pure Appl. Algebra 210 (2007) 299-306.
Date received: June 6, 2008
Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caxi-26.