Invariance and Structural Dependence

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.