Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures

Dear Colleagues,

Symmetry plays a fundamental role in our daily lives and in the study of the structure of different objects in physics, chemistry, biology, mathematics, architecture, arts, sociology, linguistics, etc. For example, the structure of molecules is well explained by their symmetry properties, described by symmetry elements and symmetry operations. A symmetry operation is a change, a transformation after which certain objects remain invariant, such as rotations, reflections, inversions, or permutation operations. Until now, the most efficient way to better describe symmetry, is using mathematical tools offered by group theory.

Naturally generalizing the concept of a group, by considering the result of the “interaction” between two elements of a non-empty set to be a set of elements (and not only one element, as for groups), Frederic Marty, in 1934, only 23 years old, defined the concept of a hypergroup. The law characterizing such a structure is called multi-valued operation, or hyperoperation, or hypercomposition and the theory of the algebraic structures endowed with at least one multi-valued operation is known as the Hyperstructure Theory or Hypercompositional Algebra. Marty’s motivation to introduce hypergroups is that the quotient of a group modulo any subgroup (not necessarily normal) is a hypergroup.

The main aim of this Special Issue is to underline various aspects in Hypercompositional Algebra, the crisp and fuzzy one, where symmetry plays a crucial role. They are related (but not limited) to equivalence relations, orderings, permutations, symmetrical groups, graphs, lattices, fuzzy sets, representations, etc. Applications of algebraic hypercompositional structures in physics, chemistry, biology, social sciences, information technologies, computer science, etc, where symmetry, or the lack of symmetry, is clearly specified and laid out, are also welcome.

Prof. Dr. Irina Cristea
Guest Editor

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