A Survey on the Mathematical Foundations of Axiomatic Entropy: Representability and Orderings

Different abstract versions of entropy, encountered in science, are interpreted in the light of numerical representations of several ordered structures, as total-preorders, interval-orders and semiorders. Intransitivities, other aspects of entropy as competitive systems, additivity, etc., are also viewed in terms of representability of algebraic structures endowed with some compatible ordering. A particular attention is paid to the problem of the construction of an entropy function or their mathematical equivalents. Multidisciplinary comparisons to other similar frameworks are also discussed, pointing out the mathematical foundations.

[1]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[2]  Esteban Induráin,et al.  Expected utility from additive utility on semigroups , 2002 .

[3]  Michele Campisi,et al.  Increase of Boltzmann entropy in a quantum forced harmonic oscillator. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Jean-Paul Doignon,et al.  On realizable biorders and the biorder dimension of a relation , 1984 .

[5]  Jesús Mosterín.  Conceptos y teorías en la ciencia , 1984 .

[6]  Eric W. Holman,et al.  Strong and weak extensive measurement , 1969 .

[7]  P. Landsberg.  Main ideas in the axiomatics of thermodynamics , 1970 .

[8]  Walter Trockel,et al.  Continuous linear representability of binary relations , 1995 .

[9]  G. Bosi,et al.  Representing preferences with nontransitive indifference by a single real-valued function☆ , 1995 .

[10]  M. Born.  Natural Philosophy of Cause and Chance , 1949 .

[11]  Esteban Induráin,et al.  Utility and entropy , 2001 .

[12]  Juan Carlos Candeal-Haro,et al.  Utility representations from the concept of measure: A corrigendum , 1994 .

[13]  Gian Paolo Beretta,et al.  Recent Progress in the Definition of Thermodynamic Entropy , 2014, Entropy.

[14]  A. Beardon,et al.  The non-existence of a utility function and the structure of non-representable preference relations , 2002 .

[15]  E. Jaynes.  Information Theory and Statistical Mechanics , 1957 .

[16]  S. Angus,et al.  Foundations of Thermodynamics , 1958, Nature.

[17]  A. Klimenko.  Complex competitive systems and competitive thermodynamics , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[18]  Juan Carlos Candeal,et al.  Existence of additive utility on positive semigroups:An elementary proof , 1998, Ann. Oper. Res..

[19]  N. Kehayopulu.  ON ORDERED Γ-SEMIGROUPS , 2010 .

[20]  Alan F. Beardon.  Debreu's Gap Theorem , 1992 .

[21]  J. Gibbs.  Elementary Principles in Statistical Mechanics , 1902 .

[22]  Juan Carlos Candeal,et al.  Scott-Suppes representability of semiorders: Internal conditions , 2009, Math. Soc. Sci..

[23]  C. Chehata.  On An Ordered Semigroup , 1953 .

[24]  Juan Carlos Candeal,et al.  Lexicographic behaviour of chains , 1999 .

[25]  Patrick Suppes,et al.  Foundational aspects of theories of measurement , 1958, Journal of Symbolic Logic.

[26]  J. Pérez-Landazábal,et al.  Reversible and irreversible martensitic transformations in Fe-Pd and Fe-Pd-Co alloys , 2008 .

[27]  Esteban Induráin,et al.  Numerical Representation of Semiorders , 2013, Order.

[28]  Françoise Point,et al.  Essentially periodic ordered groups , 2000, Ann. Pure Appl. Log..

[29]  Heinz J. Skala,et al.  Non-Archimedean Utility Theory , 1975 .

[30]  E. Induráin,et al.  Existence of Additive and Continuous Utility Functions on Ordered Semigroups , 1999 .

[31]  Kevin H. Knuth,et al.  Foundations of Inference , 2010, Axioms.

[32]  Juan Carlos Candeal,et al.  Order preserving functions on ordered topological vector spaces , 1999, Bulletin of the Australian Mathematical Society.

[33]  E. Lieb,et al.  The physics and mathematics of the second law of thermodynamics (Physics Reports 310 (1999) 1–96)☆ , 1997, cond-mat/9708200.

[34]  E. Induráin,et al.  Semiorders and thresholds of utility discrimination: Solving the Scott–Suppes representability problem , 2010 .

[35]  Peter C. Fishburn,et al.  Intransitive Indifference in Preference Theory: A Survey , 1970, Oper. Res..

[36]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[37]  Lynn Arthur Steen,et al.  Counterexamples in Topology , 1970 .

[38]  E. A. Guggenheim.  Modern thermodynamics by the methods of Willard Gibbs , 1933 .

[39]  R. Luce.  Semiorders and a Theory of Utility Discrimination , 1956 .

[40]  A. B. Kahn,et al.  Topological sorting of large networks , 1962, CACM.

[41]  Irrational ordered groups , 1988 .

[42]  R. J. Field.  Limit cycle oscillations in the reversible Oregonator , 1975 .

[43]  Juan Carlos Candeal,et al.  Order representability in groups and vector spaces , 2012 .

[44]  E. Induráin,et al.  Universal Semigroups in Additive Utility , 1998 .

[45]  Krzysztof Ciesielski.  Set Theory for the Working Mathematician: Index , 1997 .

[46]  Alexander Y. Klimenko,et al.  Complexity and intransitivity in technological development , 2014 .

[47]  Martin Oliver Steinhauser,et al.  Computational Multiscale Modeling of Fluids and Solids: Theory and Applications , 2007 .

[48]  G. Mehta.  The Euclidean Distance Approach to Continuous Utility Functions , 1991 .

[49]  Esteban Induráin,et al.  Unified Representability of Total Preorders and Interval Orders through a Single Function: The Lattice Approach , 2009, Order.

[50]  Alexander Y. Klimenko,et al.  Intransitivity in Theory and in the Real World , 2015, Entropy.

[51]  J. Neumann.  Mathematische grundlagen der Quantenmechanik , 1935 .

[52]  D. Bouyssou,et al.  Utility Maximization, Choice and Preference , 2002 .

[53]  J. Gibbs.  On the equilibrium of heterogeneous substances , 1878, American Journal of Science and Arts.

[54]  A. H. Reis,et al.  Utility function estimation: The entropy approach , 2007, 0709.0591.

[55]  Esteban Induráin,et al.  Further Results on the continuous Representability of Semiorders , 2013, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[56]  Gianni Bosi,et al.  Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof , 2002 .

[57]  Candeal,et al.  Weak Extensive Measurement without Translation-Invariance Axioms. , 1998, Journal of mathematical psychology.

[58]  Entropy and Society: Can the Physical/ Mathematical Notions of Entropy Be Usefully Imported into the Social Sphere? , 2011 .

[59]  T. Sousa,et al.  Equilibrium econophysics: A unified formalism for neoclassical economics and equilibrium thermodynamics , 2006 .

[60]  Juan Carlos Candeal,et al.  Interval-Valued Representability of Qualitative Data: the Continuous Case , 2007, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[61]  Martin Shubik,et al.  Building theories of economic process , 2009, Complex..

[62]  L. Pogliani,et al.  Constantin Carathéodory and the axiomatic thermodynamics , 2000 .

[63]  Bart Kosko,et al.  Fuzzy entropy and conditioning , 1986, Inf. Sci..

[64]  J. W. Maluf,et al.  The Teleparallel Equivalent of General Relativity and the Gravitational Centre of Mass , 2015, 1508.02465.

[65]  G. Cantor,et al.  Beiträge zur Begründung der transfiniten Mengenlehre , 1895 .

[66]  H. J. Skala.  Nonstandard utilities and the foundation of game theory , 1974 .

[67]  G. Mehta,et al.  Open gaps, metrization and utility , 1996 .

[68]  A. Tversky.  Intransitivity of preferences. , 1969 .

[69]  W. K. BURTON.  Understanding Thermodynamics , 1966, Nature.

[70]  John Bryant,et al.  A Thermodynamic Theory of Economics , 2007 .

[71]  A. Y. Klimenko,et al.  Mixing, entropy and competition , 2012, 1305.1383.

[72]  JUAN CARLOS CANDEAL,et al.  Universal codomains to Represent Interval Orders , 2009, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[73]  Juan Carlos Candeal,et al.  Representability of binary relations through fuzzy numbers , 2006, Fuzzy Sets Syst..

[74]  Alfio Giarlotta,et al.  Pointwise Debreu Lexicographic Powers , 2009, Order.

[75]  Michal Horodecki,et al.  The second laws of quantum thermodynamics , 2013, Proceedings of the National Academy of Sciences.

[76]  G. Bosi,et al.  Continuous Utility Functions Through Scales , 2008 .

[77]  E. Induráin,et al.  Entropy of chemical processes versus numerical representability of orderings , 2016, Journal of Mathematical Chemistry.

[78]  Manfred Droste,et al.  Ordinal scales in the theory of measurement , 1987 .

[79]  J. Chipman.  The Foundations of Utility , 1960 .

[80]  Jayme Luiz Szwarcfiter,et al.  A Structured Program to Generate all Topological Sorting Arrangements , 1974, Information Processing Letters.

[81]  G. Herden.  On the existence of utility functions ii , 1989 .

[82]  Kevin H. Knuth,et al.  Deriving Laws from Ordering Relations , 2004, physics/0403031.

[83]  E. Lieb,et al.  A Guide to Entropy and the Second Law of Thermodynamics , 1998, math-ph/9805005.

[84]  Marcel K. Richter,et al.  Additive utility , 1991 .

[85]  G. Debreu.  ON THE CONTINUITY PROPERTIES OF PARETIAN UTILITY , 1963 .

[86]  Eric Mayer.  A Revision Of Demand Theory , 2016 .

[87]  Matthias Dehmer,et al.  Information processing in complex networks: Graph entropy and information functionals , 2008, Appl. Math. Comput..

[88]  G. Debreu.  Mathematical Economics: Representation of a preference ordering by a numerical function , 1983 .

[89]  P. Swistak.  Some representation problems for semiorders , 1980 .

[90]  Juan Carlos Candeal,et al.  Topological Additively Representable Semigroups , 1997 .

[91]  K. Berg.  Independence and additive entropy , 1975 .

[92]  Alexander Y. Klimenko,et al.  What is mixing and can it be complex? , 2013 .

[93]  L. Fuchs.  Partially Ordered Algebraic Systems , 2011 .

[94]  V. Pokrovskii.  Econodynamics: The Theory of Social Production , 2011 .

[95]  Eric Smith,et al.  Classical thermodynamics and economic general equilibrium theory , 2008 .

[96]  On a theorem of Cooper , 2001 .

[97]  P. Fishburn.  Interval representations for interval orders and semiorders , 1973 .

[98]  A. M. W. Glass,et al.  Partially Ordered Groups , 1999 .

[99]  S. Eilenberg.  Ordered Topological Spaces , 1941 .

[100]  Juan Carlos Candeal,et al.  Archimedeaness and additive utility on totally ordered semigroups , 1996 .

[101]  C. Carathéodory.  Untersuchungen über die Grundlagen der Thermodynamik , 1909 .

[102]  Attilio Meucci,et al.  Fully Flexible Views: Theory and Practice , 2008, 1012.2848.

[103]  Marek Sawerwain,et al.  Sorting of Quantum States with Respect to Amount of Entanglement Included , 2009, CN.

[104]  Sumiyoshi Abe,et al.  Similarity between quantum mechanics and thermodynamics: entropy, temperature, and Carnot cycle. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[105]  M. Nielsen.  Conditions for a Class of Entanglement Transformations , 1998, quant-ph/9811053.

[106]  Juan Carlos Candeal-Haro,et al.  UTILITY FUNCTIONS ON PARTIALLY ORDERED TOPOLOGICAL GROUPS , 1992 .

[107]  Asier Estevan.  Generalized Debreu’s Open Gap Lemma and Continuous Representability of Biorders , 2016, Order.

[108]  Juan Carlos Candeal-Haro,et al.  Utility representations from the concept of measure , 1993 .

[109]  Juan Carlos Candeal-Haro,et al.  A note on linear utility , 1995 .

[110]  T. Sousa,et al.  Is neoclassical microeconomics formally valid? An approach based on an analogy with equilibrium thermodynamics , 2006 .

[111]  G. Cantor,et al.  Beiträge zur Begründung der transfiniten Mengenlehre. (Zweiter Artikel.) , 2022 .

[112]  Juan Carlos Candeal,et al.  Continuous order representability properties of topological spaces and algebraic structures , 2012 .

[113]  J. Milewska-Duda,et al.  A theoretical model for evaluation of configurational entropy of mixing with respect to shape and size of particles , 1995 .

[114]  J. Falmagne.  A Set of Independent Axioms for Positive Holder Systems , 1975, Philosophy of Science.

[115]  T. Rader,et al.  The Existence of a Utility Function to Represent Preferences , 1963 .

[116]  Alexander Y. Klimenko,et al.  Entropy and Equilibria in Competitive Systems , 2013, Entropy.

[117]  A remark on a utility representation theorem of Rader , 1997 .

[118]  Juan Carlos Candeal,et al.  Numerical Representability of Ordered Topological Spaces with Compatible Algebraic Structure , 2011, Order.

[119]  Juan Carlos Candeal,et al.  Numerical representability of semiorders , 2002, Math. Soc. Sci..

[120]  R. Santamarta,et al.  Entropy change linked to the magnetic field induced Morin transition in Hematite nanoparticles , 2012 .

[121]  D. Bridges,et al.  Representations of Preferences Orderings , 1995 .

[122]  R. Clausius,et al.  Ueber die Wärmeleitung gasförmiger Körper , 1862 .

[123]  E. Induráin,et al.  Extensive Measurement: Continuous Additive Utility Functions on Semigroups , 1996 .

[124]  P. Fishburn.  Intransitive indifference with unequal indifference intervals , 1970 .

[125]  I. Gilboa,et al.  Numerical representations of imperfectly ordered preferences (a unified geometric exposition) , 1992 .

[126]  Robert Bowen.  A New Proof of a Theorem in Utility Theory , 1968 .

[127]  C. Adami,et al.  Negative entropy and information in quantum mechanics , 1995, quant-ph/9512022.

[128]  Juan Carlos Candeal,et al.  Numerical Representations of Interval Orders , 2001, Order.

[129]  Juan Carlos Candeal,et al.  Isotonies on ordered cones through the concept of a decreasing scale , 2007, Math. Soc. Sci..

[130]  Kenneth L. Manders.  On jnd representations of semiorders , 1981 .

[131]  Angus Macintyre,et al.  Free abelian lattice-ordered groups , 2005, Ann. Pure Appl. Log..

[132]  Rongxi Zhou,et al.  Applications of Entropy in Finance: A Review , 2013, Entropy.

[133]  Javier Gutiérrez García,et al.  Semiorders with separability properties , 2012 .

[134]  Fred S. Roberts,et al.  Axiomatic thermodynamics and extensive measurement , 1968, Synthese.

[135]  S. Gensemer.  On numerical representations of semiorders , 1988 .

[136]  Philippe Vincke,et al.  Semiorders - Properties, Representations, Applications , 1997, Theory and decision library: series B.

[137]  Juan Carlos Candeal,et al.  Representations of ordered semigroups and the Physical concept of Entropy , 2004 .

[138]  J. Jaffray.  Existence of a Continuous Utility Function: An Elementary Proof , 1975 .

[139]  Esteban Induráin,et al.  Continuous Representability of Interval Orders: The Topological Compatibility Setting , 2015, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[140]  R. Clausius,et al.  Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie , 1865 .