Semi-algebraic function rings and reflectors of partially ordered rings
type: Item
instance of: scholarly article
title: Semi-algebraic function rings and reflectors of partially ordered rings
zbMATH Open document ID: 0967.14038
author: Niels Schwartz
author: J. Madden
published in: Lecture Notes in Mathematics
publication date: 1999-11-07T00:00:00Z
review text: The book under review develops algebraic techniques which are especially adapted to the requirements of real algebraic geometry. A lot of function rings appear in real algebraic geometry as extensions (a sort of completions or closures) of the ring of polynomial functions on an algebraic set. The present work shows how these extensions can be described and treated in a general framework of monoreflectors of the category of reduced partially ordered rings. A subcategory \({\mathbf D}\) of a category \({\mathbf C}\) is reflective if the inclusion functor has a left adjoint \(r:{\mathbf C} \to{\mathbf D}\). In this case \(r\) is called a reflector, and \(r_A:A\to r(A)\) (where \(A\) is an object of \({\mathbf C})\) is called the reflection morphism of \(A\). A reflector is a monoreflector if the reflection morphisms are monomorphisms. The monoreflective subcategories of the category POR/N of reduced partially ordered rings form a complete lattice. An important class of reflectors is composed by the \(H\)-closed monoreflectors. A monoreflector \(r:\text{POR/N} \to{\mathbf D}\) on the category of reduced partially ordered rings is called \(H\)-closed if for any partially ordered ring \((A,P)\) belonging to \({\mathbf D}\) and any surjective homomorphism \((A,P) \to(B,Q)\) in POR/N, the partially ordered ring \((B,Q)\) also belongs to \({\mathbf D}\). The \(H\)-closed monoreflective subcategories of POR/N also form a complete lattice. The book contains 23 sections. Sections 1-5 are introductory. Sections 6-15 are devoted to rings of semi-algebraic functions and real closed rings. A systematic study of some intervals in the lattice of \(H\)-closed monoreflectors is performed in sections 16-22. Section 23 contains a summary of results on the lattice of monoreflectors of the category POR/N of reduced partially ordered rings and on the lattice of \(H\)-closed monoreflectors of POR/N.
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 18-02
Mathematics Subject Classification ID: 06-02
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 13J25
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 18B35
zbMATH DE Number: 1357591
zbMATH Keywords: real algebraic geometry
zbMATH Keywords: complete lattice
reviewed by: Ilia Itenberg
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0093968
OpenAlex ID: W582059137
DOI: 10.1007/BFB0093968
description: scientific article; zbMATH DE number 1357591