Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Mon, 22 Jul 2019 05:44:09 GMT2019-07-22T05:44:09Z50161Notes on point-free real functions and sublocaleshttp://hdl.handle.net/10316/43900Title: Notes on point-free real functions and sublocales
Authors: Gutiérrez García, Javier; Picado, Jorge; Pultr, Aleš
Abstract: Using the technique of sublocales we present a survey of some known facts (with a few new ones added) on point-free real functions. The subjects treated are, e.g., images and preimages, semicontinuity, algebraic structure (point-free real arithmetics), zero and cozero parts, z-embeddings, z-open and z-closed maps, disconnectivity, small sublocales and supports.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10316/439002014-01-01T00:00:00ZOn the parallel between normality and extremal disconnectednesshttp://hdl.handle.net/10316/43789Title: On the parallel between normality and extremal disconnectedness
Authors: Gutiérrez García, Javier; Picado, Jorge
Abstract: Several familiar results about normal and extremally disconnected (classical or pointfree) spaces shape the idea that the two notions are somehow dual to each other and can therefore be studied in parallel. This paper investigates the source of this ‘duality’ and shows that each pair of parallel results can be framed by the ‘same’ proof. The key tools for this purpose are relative notions of normality, extremal disconnectedness, semicontinuity and continuity (with respect to a fixed class of complemented sublocales of the given locale) that bring and extend to locale theory a variety of well-known classical variants of normality and upper and lower semicontinuities in an illuminating unified manner. This approach allows us to unify under a single localic proof all classical insertion, as well as their corresponding extension results.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10316/437892014-01-01T00:00:00ZNormal semicontinuity and the Dedekind completion of pointfree function ringshttp://hdl.handle.net/10316/43797Title: Normal semicontinuity and the Dedekind completion of pointfree function rings
Authors: Gutiérrez García, Javier; Mozo Carollo, Imanol; Picado, Jorge
Abstract: This paper supplements an earlier one by the authors which constructed the Dedekind completion of the ring of continuous real functions on an arbitrary frame L in terms of partial continuous real functions on L. In the present paper, we provide three alternative views of it, in terms of (i) normal semicontinuous real functions on L, (ii) the Booleanization of L (in the case of bounded real functions) and the Gleason cover of L (in the general case), and (iii) Hausdorff continuous partial real functions on L. The first is the normal completion and extends Dilworth’s classical construction to the pointfree setting. The second shows that in the bounded case, the Dedekind completion is isomorphic to the lattice of bounded continuous real functions on the Booleanization of L, and that in the non-bounded case, it is isomorphic to the lattice of continuous real functions on the Gleason cover of L. Finally, the third is the pointfree version of Anguelov’s approach in terms of interval-valued functions. Two new classes of frames, cb-frames and weak cb-frames, emerge naturally in the first two representations. We show that they are conservative generalizations of their classical counterparts.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10316/437972016-01-01T00:00:00ZMonotone normality and stratifiability from a pointfree point of viewhttp://hdl.handle.net/10316/43791Title: Monotone normality and stratifiability from a pointfree point of view
Authors: Gutiérrez García, Javier; Picado, Jorge; de Prada Vicente, María Ángeles
Abstract: Monotone normality is usually defined in the class of T_1 spaces. In this paper we study it under the weaker condition of subfitness, a separation condition that originates in pointfree topology. In particular, we extend some well known characterizations of these spaces to the subfit context (notably, their hereditary property and the preservation under surjective continuous closed maps) and present a similar study for stratifiable spaces, an important subclass of monotonically normal spaces. In the second part of the paper, we extend further these ideas to the lattice theoretic setting. In particular, we give the pointfree analogues of the previous results on monotonically normal spaces and introduce and investigate the natural pointfree counterpart of stratifiable spaces.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10316/437912014-01-01T00:00:00ZPresenting the frame of the unit circlehttp://hdl.handle.net/10316/44424Title: Presenting the frame of the unit circle
Authors: Gutiérrez García, Javier; Mozo Carollo, Imanol; Picado, Jorge
Abstract: We present the frame L(T) of the unit circle by generators and relations in two alternative ways. The first is the localic counterpart of the Alexandroff compactification of the real line while the other can be understood as a localic analogue of the quotient space R/Z. With an eye towards a prospective point-free description of Pontryagin duality, we then show how the usual group operations of the frame of reals can be lifted to the new frame L(T), endowing it with a canonical localic group structure.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10316/444242016-01-01T00:00:00ZA unified view of the Dedekind completion of pointfree function ringshttp://hdl.handle.net/10316/43890Title: A unified view of the Dedekind completion of pointfree function rings
Authors: Gutiérrez García, Javier; Mozo Carollo, Imanol; Picado, Jorge
Abstract: We provide the appropriate unifying framework for the various descriptions of the Dedekind completion of the ring C(L) of continuous real functions on a frame L. It is based on suitable Galois connections and a general result about Galois connections, showing once more the ubiquity of (Galois) adjunctions between partially ordered sets and their conceptual simplicity and eﬀectiveness.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10316/438902016-01-01T00:00:00ZPerfectness in localeshttp://hdl.handle.net/10316/43888Title: Perfectness in locales
Authors: Gutiérrez García, Javier; Kubiak, Tomasz; Picado, Jorge
Abstract: This paper makes a comparison between two notions of perfectness for locales which come as direct reformulations of the two equivalent topological definitions of perfectness. These reformulations are no longer equivalent. It will be documented that a locale may appropriately be called perfect if each of its open sublocales is a join of countably many closed sublocales. Certain circumstances are exhibited in which both reformulations coincide. This paper also studies perfectness in mildly normal locales. It is shown that perfect and mildly normal locales coincide with the Oz locales extensively studied in the last decade.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10316/438882017-01-01T00:00:00ZOn the Dedekind completion of function ringshttp://hdl.handle.net/10316/43901Title: On the Dedekind completion of function rings
Authors: Mozo Carollo, Imanol; Gutiérrez García, Javier; Picado, Jorge
Abstract: This paper introduces the frame of partially defined real numbers and the lattice-ordered ring of partial real functions on a frame. This is then used to construct the order completion of rings of pointfree continuous real functions. The bounded and integer-valued cases are also analysed. The application of this pointfree approach to the classical case C(X) of the ring of continuous real-valued functions on a topological space X yields a new construction for the Dedekind completion of C(X), considerably more direct and natural than the known procedure using Hausdorff continuous functions.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10316/439012015-01-01T00:00:00ZMonotone insertion and monotone extension of frame homomorphismshttp://hdl.handle.net/10316/4588Title: Monotone insertion and monotone extension of frame homomorphisms
Authors: Gutiérrez García, Javier; Kubiak, Tomasz; Picado, Jorge
Abstract: The purpose of this paper is to introduce monotonization in the setting of pointfree topology. More specifically, monotonically normal locales are characterized in terms of monotone insertion and monotone extensions theorems.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/45882008-01-01T00:00:00ZLocalic real functions: A general settinghttp://hdl.handle.net/10316/10042Title: Localic real functions: A general setting
Authors: Gutiérrez García, Javier; Kubiak, Tomasz; Picado, Jorge
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10316/100422009-01-01T00:00:00ZUniform-type structures on lattice-valued spaces and frameshttp://hdl.handle.net/10316/4595Title: Uniform-type structures on lattice-valued spaces and frames
Authors: Gutiérrez García, Javier; Mardones-Pérez, Iraide; Picado, Jorge; Prada Vicente, María Angeles de
Abstract: By introducing lattice-valued covers of a set, we present a general framework for uniform structures on very general L-valued spaces (for L an integral commutative quantale). By showing, via an intermediate L-valued structure of uniformity, how filters of covers may describe the uniform operators of Hutton, we prove that, when restricted to Girard quantales, this general framework captures a significant class of Hutton's uniform spaces. The categories ofL-valued uniform spaces and L-valued uniform frames here introduced provide (in the case L is a complete chain) the missing vertices in the commutative cube formed by the classical categories of topological and uniform spaces and their corresponding pointfree counterparts (forming the base of the cube) and the corresponding L-valued categories (forming the top of the cube).
Mon, 01 Sep 2008 11:34:57 GMThttp://hdl.handle.net/10316/45952008-09-01T11:34:57ZPointfree forms of Dowker and Michael insertion theoremshttp://hdl.handle.net/10316/11276Title: Pointfree forms of Dowker and Michael insertion theorems
Authors: Gutiérrez García, Javier; Kubiak, Tomasz; Picado, Jorge
Abstract: In this paper we prove two strict insertion theorems for frame homomorphisms.
When applied to the frame of all open subsets of a topological space
they are equivalent to the insertion statements of the classical theorems of Dowker
and Michael regarding, respectively, normal countably paracompact spaces and perfectly
normal spaces. In addition, a study of perfect normality for frames is made.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10316/112762007-01-01T00:00:00ZLower and upper regularizations of frame semicontinuous real functionshttp://hdl.handle.net/10316/11293Title: Lower and upper regularizations of frame semicontinuous real functions
Authors: Gutiérrez García, Javier; Kubiak, Tomasz; Picado, Jorge
Abstract: As discovered recently, Li andWang's 1997 treatment of semicontinuity
for frames does not faithfully re
ect the classical concept. In this paper we continue
our study of semicontinuity in the pointfree setting. We de ne the pointfree
concepts of lower and upper regularizations of frame semicontinuous real functions.
We present characterizations of extremally disconnected frames in terms of these
regularizations that allow us to reprove, in particular, the insertion and extension
type characterizations of extremally disconnected frames due to Y.-M. Li and Z.-H.
Li [Algebra Universalis 44 (2000), 271{281] in the right semicontinuity context. It
turns out that the proof of the insertion theorem becomes very easy after having
established a number of basic results regarding the regularizations. Notably, our
extension theorem is a much strengthened version of Li and Li's result and it is
proved without making use of the insertion theorem.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10316/112932007-01-01T00:00:00ZInsertion of continuous real functions on spaces, bispaces, ordered spaces and point-free spaces - a common roothttp://hdl.handle.net/10316/11191Title: Insertion of continuous real functions on spaces, bispaces, ordered spaces and point-free spaces - a common root
Authors: Ferreira, Maria João; Gutiérrez García, Javier; Picado, Jorge
Abstract: We characterize normal and extremally disconnected biframes in terms
of the insertion of a continuous real function in between given lower and upper semicontinuous
real functions and show this to be the common root of several classical
and new insertion results concerning topological spaces, bitopological spaces, ordered
topological spaces and locales.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10316/111912009-01-01T00:00:00ZRings of real functions in Pointfree Topologyhttp://hdl.handle.net/10316/13708Title: Rings of real functions in Pointfree Topology
Authors: Gutiérrez García, Javier; Picado, Jorge
Abstract: This paper deals with the algebra F(L) of real functions of a frame L
and its subclasses LSC(L) and USC(L) of, respectively, lower and upper semicontinuous
real functions. It is well-known that F(L) is a lattice-ordered ring; this paper
presents explicit formulas for its algebraic operations which allow to conclude about
their behaviour in LSC(L) and USC(L).
As applications, idempotent functions are characterized and the results of [10]
about strict insertion of functions are signi cantly improved: general pointfree formulations
that correspond exactly to the classical strict insertion results of Dowker
and Michael regarding, respectively, normal countably paracompact spaces and perfectly
normal spaces are derived.
The paper ends with a brief discussion concerning the frames in which every
arbitrary real function on the -dissolution of the frame is continuous
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10316/137082010-01-01T00:00:00ZCompletely normal frames and real-valued functionshttp://hdl.handle.net/10316/11233Title: Completely normal frames and real-valued functions
Authors: Ferreira, Maria João; Gutiérrez García, Javier; Picado, Jorge
Abstract: Up to now point-free insertion results have been obtained only for
semicontinuous real functions. Notably, there is now available a setting for dealing
with arbitrary, not necessarily (semi-)continuous, point-free real functions, due to
Guti errez Garc a, Kubiak and Picado, that gives point-free topology the freedom
to deal with general real functions only available before to point-set topology. As a
rst example of the usefulness of that setting, we apply it to characterize completely
normal frames in terms of an insertion result for general real functions. This characterization
extends a well known classical result of T. Kubiak about completely
normal spaces. In addition, characterizations of completely normal frames that extend
results of H. Simmons for topological spaces are presented. In particular, it
follows that complete normality is a lattice-invariant property of spaces, correcting
an erroneous conclusion in [Y.-M. Wong, Lattice-invariant properties of topological
spaces, Proc. Amer. Math. Soc. 26 (1970) 206-208].
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/112332008-01-01T00:00:00Z