Numerical representability of ordered topological spaces with compatible algebraic structure
This paper is devoted to the study of continuous numerical representations of totally preordered topological vector spaces through real-valued order-preserving functions that are also algebraic homomorphisms. The main novelty is the analysis of numerical representations that are continuous, linear and multiplicative, motivated from applications in Economics and Social Sciences. For this the authors endow totally preordered topological vector spaces with an additional binary relation, compatible with the given preorder, obtaining the category of totally preordered semitopological real algebras. They provide conditions that characterize the existence of continuous real-valued order-preserving maps that are linear and multiplicative and represent total preorders defined on a semitopological real algebra. Then they present a number of results inspired by important concepts and issues arising in Economics, Social Choice and Decision-Making theories (like the problem of aggregating individual judgements or opinions into a social one). There is a flaw in Theorem 3.14 due to a condition missing in Definition 3.5. This has recently been corrected by the authors in an erratum [Order 29, No. 1, 227--227 (2012; Zbl 1236.54029)].
continuous numerical representations of total preorders
scientific article; zbMATH DE number 6018075
scientific article