Two functors induced by certain ideals of function rings.
type: Item
instance of: scholarly article
title: Two functors induced by certain ideals of function rings.
zbMATH Open document ID: 1306.06007
author: Themba Dube
author: Oghenetega Ighedo
published in: Applied Categorical Structures
publication date: 2014-09-19T00:00:00Z
review text: Let \(L\) be a completely regular frame and let \(\mathcal RL\) be the ring of continuous real-valued functions on \(L\). This paper deals with the lattices of \(z\)-ideals and \(d\)-ideals of \(\mathcal RL\), respectively, denoted by \(\mathrm{Zid}(\mathcal RL)\) and \(\mathrm{Did}(\mathcal RL)\). In previous papers the authors proved that \(\mathrm{Zid}(\mathcal RL)\) and \(\mathrm{Did}(\mathcal RL)\) are both coherent frames. In the present paper they show that the assignments \(L\mapsto\mathrm{Zid}(\mathcal RL)\) and \(L\mapsto\mathrm{Did}(\mathcal RL)\) are functorial and construct a natural transformation between the corresponding functors \(\mathsf Z\colon\mathsf{CRegFrm}\to\mathsf{CohFrm}\) and \(\mathsf D\colon\mathsf{CRegFrm}\to\mathsf{CohFrm}\). Then they prove that both \(\mathsf Z\) and \(\mathsf D\) preserve and reflect skeletal and \(*\)-dense frame homomorphisms. Finally they show that, inside the collection of \(\lambda\)-maps, \(\mathsf Z\) also preserves and reflects the property of a frame map to have a left adjoint. It follows, in particular, that for \(\lambda\)-maps \(\mathsf Z\) preserves and reflects open maps.
reviewed by: Jorge Picado
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 18A22
Mathematics Subject Classification ID: 54C30
zbMATH DE Number: 6346476
zbMATH Keywords: coherent frames
zbMATH Keywords: \(z\)-ideals
zbMATH Keywords: \(d\)-ideals
zbMATH Keywords: faithful functors
zbMATH Keywords: \(*\)-density
zbMATH Keywords: skeletal maps
zbMATH Keywords: open maps
zbMATH Keywords: left adjoint functors
MaRDI profile type: Publication
full work available at URL: s10485-013-9320-0
OpenAlex ID: W1975315958
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DOI: 10.1007/S10485-013-9320-0
description: scientific article; zbMATH DE number 6346476