Semi-algebraic function rings and reflectors of partially ordered rings
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.
scientific article; zbMATH DE number 1357591