Syntopy groups

V. ANOTHER FUZZY SYMMETRY APPROACH SYNTOPY AND SYNTOPY GROUPS... 

Syntopy and syntopy groups were introduced in an early approach to a fuzzy set representation of approximate symmetry, where imperfect symmetry is regarded as fuzzy symmetry. Whereas any symmetry is a discrete property within a metric space, it is natural to consider a fuzzy set approach for a continuous extension of the discrete symmetry concept to quasisymmetric objects, such as some almost symmetric molecular structures. The syntopy approaches take into account the nonlocalized, quantum-mechanical, fuzzy nature of nuclear arrangements of molecules. 

Within the syntopy model, the essential algebraic structure of point symmetry groups is retained (in fact, this structure is extended), and the elements of syntopy groups are derived from ordinary point symmetry operators [252,394,395]. There are, however, alternative approaches for the generalization of symmetry, where fundamentally different algebraic structures are used. 

The discontinuous symmetry changes and the binary nature of the presence or absence of symmetry elements hinders the application of point group symmetry methods for general molecular structures. In the syntopy approach, based on fuzzy set theory, the discrete concept of point symmetry is replaced by a continuous concept and is applicable to cases of almost symmetric or quasisymmetric molecular arrangements. When replacing symmetry with syntopy, some of the advantages of the group... 

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