Abstract
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.
Highlights
Entropy Theory can be compared to several other interdisciplinary mathematical theories that involve numerical representations of ordered structures
We have selected that paper by Cooper ([18]), dated 1967, because it shows a clear formulation to compare axiomatic Thermodynamics to representability theory of ordered structures in Mathematics
This study is necessary, in our opinion, because the construction of numerical representations of ordered structures in a pure mathematical abstract setting could seriously differ from the typical understanding of an entropy in Thermodynamics as, say, a physical concept given by a suitable path-independent integral
Summary
Entropy Theory can be compared to several other interdisciplinary mathematical theories that involve numerical representations of ordered structures (see, e.g., [1]). We have selected that paper by Cooper ([18]), dated 1967, because it shows a clear formulation to compare axiomatic Thermodynamics to representability theory of ordered structures in Mathematics. We insist in this key fact that, from the point of view of mathematicians, those physical approaches have exactly the same mathematical treatment, namely the use of representability theory of ordered structures to convert qualitative scales into quantitative ones. There exist past related works on axiomatic entropy in the teleparallel alternative of general relativity with a non-minimal coupling between the gravity and scalar field (see, e.g., [30]) In some of these studies, ordering relations constitute a basis to obtain laws, as we do in the present paper.)
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