Presenting the frame of the unit circle.
type: Item
instance of: scholarly article
title: Presenting the frame of the unit circle.
zbMATH Open document ID: 1332.06027
author: Javier Gutiérrez García
author: Imanol Mozo Carollo
author: Jorge Picado
published in: Journal of Pure and Applied Algebra
publication date: 2015-11-10T00:00:00Z
full work available at URL: http://hdl.handle.net/10316/44424
review text: The frame of reals, \(\mathfrak L(\mathbb R)\), can be presented as a frame generated by pairs \((p,q)\) of all rationals subject to some relations. It can also be presented as a frame generated by abstract ``rays'' \((p,-)\) and \((-, q)\), for \(p,q\in\mathbb Q\), subject to relations that mimic, in a way, how the rays \((p,\infty)\), \((-\infty,p)\) relate. Since \(\mathfrak L(\mathbb R)\) is isomorphic to the frame of open sets of the line, it is natural to ask if the frame of open sets of the unit circle can be presented by generators and relations. Incidentally, this question was posed to the authors by Bernhard Banaschewski. In answering this, the authors present frame \(\mathfrak L(\mathbb T)\) of the unit circle by generators and relations in two alternative ways. The first is the point-free counterpart of the Alexandroff compactification of the real line; and the second can be viewed as a localic analogue of the quotient space \(\mathbb R/\mathbb Z\). They prove, among other things, that the spectrum of \(\mathfrak L(\mathbb T)\) is homeomorphic to the unit circle; which accords with the case of the frame of reals. They show how the usual group operations of \(\mathfrak L(\mathbb R)\) can be lifted to \(\mathfrak L(\mathbb T)\), thus endowing it with a canonical localic group structure. As they remark, this is forward looking to a prospective point-free description of Pontryagin duality.
reviewed by: Themba Dube
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 18D35
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54H11
zbMATH DE Number: 6506971
zbMATH Keywords: frames
zbMATH Keywords: frame of reals
zbMATH Keywords: frame of the unit circle
zbMATH Keywords: localic groups
MaRDI profile type: Publication
OpenAlex ID: W1840422603
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DOI: 10.1016/J.JPAA.2015.08.004
description: scientific article; zbMATH DE number 6506971