This book deals with the notices of structural dependenceand independence, which are used in many applications ofmathematics to science. For instance, a physical law statesthat one physical aspect is structurally dependent on one ormore other aspects. Structural dependence is closely relatedto the mathematical idea of functional dependence. However, functional dependence expresses a kind of completedependence while structural dependence includes partialdependence. Further, structural dependence is primarilythought of as a relation holding between aspects rather thanbetween their measures. In this book, the traditional way oftreating aspects within measurement theory is modified. Anaspect is not viewed as a set-theoretical structure but as afunction which has sets as arguments and set-theoreticalstructures as values. This way of regardingaspects isillustrated with an application to social choice and groupdecision theory. Structural dependence is connected with theidea of concomitant variations and the mathematical notionof invariance. A distinction between dependence in the senseof determination and in the sense of relevance is drawn.
Structural dependence both in the sense of determination andof relevance can be found in degrees from completedetermination and relevance to complete undetermination andirrelevance. The possibility of different "scales" ofdetermination and relevance is considered. One chapter ofthe book is devoted to a study of the dependence relationbetween the social ordering and the individual orderings insocial choice and group decision theory.
1. Problem Area and Basic Formal Apparatus.- 1. The Concept of Dependence in Applied Mathematics; a First Account.- 1.0 Introduction.- 1.1 Determination and relevance.- 1.2 Partial determination.- 1.3 Structural dependence.- 1.4 Dependence and concomitant variations.- 1.5 Supervenience and dependence.- 1.6 Invariance and dependence.- 1.7 Independence of primitive symbols.- 1.8 Relations as functions.- 1.9 Notions of independence in modern measurement and decision theory.- 1.10 Applications of structural dependence.- 1.11 Summing up.- 2. Basic Formal Concepts and Terminology.- 2.0 Introduction.- 2.1 Relations and functions.- 2.2 Properties of binary relations.- 2.3 Order relations.- 2.4 Two lemmas on weak orders.- 2.5 Semiorders.- 2.6 Correspondences.- 2.7 Invariance.- 2.8 Relational structures.- 2.9 Isomorphisms and homomorphisms.- 2.10 Congruence relations.- 2.11 Lattices.- 2. An Informal Presentation of the Main Themes.- 3. Relationals.- 3.0 Introduction.- 3.1 The fundamentals of relational.- 3.2 Formal properties of relationals.- 3.3 Some examples.- 3.4 Finitary systems of relationals.- 3.5 Historical and bibliographical remarks.- 4. Subordination, Uncorrelation and Derivation.- 4.0 Introduction.- 4.1 Isomorphism preservation and transitions.- 4.2 Subordination and definability.- 4.3 Uncorrelation.- 4.4 The dependence between R and its regionalization R*.- 4.5 Equality and decision methods for relationals.- 4.6 Derived and derivable relationals.- 4.7 Stability of transitions.- 4.8 The structural character of transitions and subordination.- 4.9 Significance.- 5. An Example: Social Choice.- 5.0 Introduction.- 5.1 The notion of dependence in social choice theory.- 5.2 Preference relationals and collective choice rules.- 5.3 Isomorphism preservation, subordination and social choice.- 5.4 Relative effectiveness, derivability and social choice.- 5.5 Stability and background for collective choice rules.- 5.6 Structural dependence and aggregation; a preliminary remark.- 6. Conformity and Measures.- 6.0 Introduction.- 6.1 Equality preservation and independent realizability.- 6.2 Congruence relational, conformity and import.- 6.3 Homomorphic representations.- 6.4 Measures.- 6.5 Numerical measures and representations.- 6.6 Connections between relational defined by measures.- 3. Formal Treatment of Basic Topics.- 7. Transitions Between Systems of Relationals.- 7.0 Introduction.- 7.1 Relational systems.- 7.2 Transitions and subordination.- 7.3 Transitions and uncorrelation.- 7.4 Concatenation and transition.- 7.5 Significance.- 7.6 Stability and monotonicity of s-functions.- 8. The Structure of Subordination.- 8.0 Introduction.- 8.1 Subalternation and rank.- 8.2 The lattice of subalternation.- 8.3 Correlation and collaterally.- 8.4 Semiranks.- 8.5 On equality preservation and independent realizability of structures.- 9. Isomorphic Mappings and Invariance.- 9.0 Introduction.- 9.1 Mappings.- 9.2 Isomorphic mappings.- 9.3 Automorphic mapping invariance.- 9.4 Global isomorphic mappings and global subordination.- Final remarks.- References.
Series: Lecture Notes in Economic and Mathematical Systems
Number Of Pages: 245
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 24.41 x 16.99
Weight (kg): 0.43