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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

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Volume 27, Issue 5


On the Dedekind completion of function rings

Imanol Mozo Carollo / Javier Gutiérrez García / Jorge Picado
Published Online: 2013-10-16 | DOI: https://doi.org/10.1515/forum-2012-0095


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.

Keywords: Frame of reals; frame of partial reals; continuous real function; partial real function; order complete; Dedekind completion; completely regular frame; extremally disconnected frame; zero-dimensional frame

MSC: 06D22; 06B23; 54C30; 26A15

About the article

Received: 2012-07-06

Revised: 2013-05-27

Published Online: 2013-10-16

Published in Print: 2015-09-01

Funding Source: Ministry of Economy and Competitiveness of Spain

Award identifier / Grant number: MTM2012-37894-C02-02

Funding Source: UPV/EHU

Award identifier / Grant number: UFI11/52

Funding Source: UPV/EHU

Award identifier / Grant number: GIU12/39

Funding Source: Centre for Mathematics of the University of Coimbra

Award identifier / Grant number: PEst-C/MAT/UI0324/2011

Citation Information: Forum Mathematicum, Volume 27, Issue 5, Pages 2551–2585, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2012-0095.

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