Special Issue on Symmetry in Classical and Fuzzy Algebraic Hypercompositional Structures

1. Introduction

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 was using mathematical tools offered by group theory, but recently, new methods combining elements from different theories, for example graph theory, hypercompositional algebra, or ordered algebra, have been developed in order to cope with symmetries.

Hypercompositional algebra is a well established branch of abstract algebra, born in 1934 when the French mathematician Frederic Marty introduced the concept of a hypergroup during the 8th Congress of Scandinavian Mathematicians [1]. Hypercompositional algebra deals with structures endowed with multi-valued operations, called hyperoperations or hypercompositions. These are natural generalizations of classical operations with the property that the result of the hyperoperation is a subset of the carrier set, instead of a single element, as it happens in the classical algebraic structures endowed with operations. The algebraic structures endowed with multivalued operations currently have broad applications in many areas of mathematics—for example, multivalued formal groups have important applications in algebraic topology, multivalued Lie groups in functional equations and integrable systems, join spaces in geometry, etc., but also in physics, chemistry, biology, and social sciences. It is worth mentioning here the contributions of Alain Connes, winner of the Fields medal, in the theory of algebraic curves related to the theory of hyperfields [2,3]. These days, hypercompositional algebra has proved to be an excellent tool in modeling and solving crucial problems in algebraic geometry [4], number theory [5], affine algebraic group schemes [6], and tropical geometry or supertropical algebras [7].

2. Contributions

The main aim of this Special Issue is to underline various aspects in hypercompositional algebra, the crisp [8,9] and fuzzy one [10], where symmetry plays a crucial role. It is a collection of 12 innovative research papers in the field of hypercompositional algebra, 7 of them being more theoretically oriented, while the other 5 present also strong applicative aspects in engineering, control theory, artificial intelligence, or graph theory. Their main findings are summarized in the following, while the interested readers can directly consult the papers for more details.

The theoretical aspects discussed in the articles of this book are principally related with three main topics: (semi)hypergroups, hyperfields and BCK-algebras. Heidari and Cristea [11] present a natural generalization of breakable semigroups, defining a new hypercompositional structure called a breakable semihypergroup, where every non-empty subset is a subsemihypergroup. The authors proved that a hypergroup is breakable if and only if it is a B-hypergroup. Moreover they obtained a generalization of Redei’s theorem for semigroups, which gives a decomposition of semi-symmetric semihypergroups in classes that are ordered in such a way that each class is an l-semigroup, an r-semigroup or a B-hypergroup. Notice that this theorem covers also the case of all algebraic semigroups, not included in the original Redei’s theorem.

Using the fundamental relation $\beta$ on a hypergroup, the smallest strongly regular equivalence defined on a hypergroup such that the corresponding quotient structure is a group, some new properties of the $\beta$-classes are obtained by De Salvo et al. [12]. The authors introduced and investigated the notion of height of a $\beta$-class, helping to characterize the 1-hypergroups and to define some new hypergroup classes, based on the height and the cardinality of the $\beta$-classes.

Based on the properties of a cyclic hypergroup of particular matrices, Krehlik and Vyroubalova described in [13] the symmetry of lower and upper approximations in certain rough sets connected with this hypergroup. The main tool involved in their study is the Ends lemma, in which the hypercomposition is the principal end of a partially ordered semigroup. Their findings suggest an application for the study of the detection sensors, used in mobile robot mapping.

In the framework of hyperrings and hyperfields theory, new lines of research have been developed regarding the hyperhomographies on Krasner hyperfields, with interesting applications in cryptography [14], and new fuzzy weak hyperideals have been defined in $H_v$-rings by using the concept of fuzzy multiset [15]. In particular, Vahedi et al. [14] investigated the properties of a hypergroup obtained by extending a homography $H_{a,b}\left(F\right)=\{(x,y)\in F^2\mid y=b+\frac{1}{x-a}\}$ on a field F to a hyperhomography on the quotient Krasner hyperfield associated with F. This construction may find applications in cryptography in connection with the Berardi’s cryptographic system.

Al Tahan et al. [15] defined for the first time some algebraic hypercompositional structures related with fuzzy multisets, obtaining the so-called fuzzy multi-$H_v$-ideals of $H_v$-rings. We recall here that the $H_v$-rings are multivalued systems endowed with addition and multipication (both being hyperoperations) connected by the weak distributivity property.

Two articles are dedicated to the study of BCK-algebras. Bordbar et al. [16] present properties of the relative annihilators in lower $BCK$-semilattices. More precisely, the authors give necessary conditions for a relative annihilator in a lower $BCK$-algebra to be a prime ideal. After defining the minimal prime decomposition of an ideal of a $BCK$-algebra, the authors investigated the homomorphic image and the pre-image of such a decomposition. These results help them to study new properties of the semi-prime closure operation “$cl$” on a $BCK$-algebra.

Another approach to the theory of $BCK$-algebras is presented in the paper [17] by Xin et al., where several types of intuitionistic fuzzy soft ideals in hyper $BCK$-algebras are defined and studied. Here, the authors combined the theory of intuitionistic fuzzy soft sets with the one of hyper $BCK$-algebra, investigating the properties of several types of hyperideals.

Recently, more and more researchers have been interested in the applicative aspects of the algebraic hypercompositional structures and this Special Issue covers some of them. Some applications of operations defined on multisets are covered by [18]. Inspired by the use of structures of linear differential operators, Chvalina and Smetana [19] defined and studied the group and the hypergroup related to the set of artificial neurons, emphasizing their symmetrical properties.

In Novak et al. [20], a mathematical model, based on elements of algebraic hyperstructure theory used in the context of underwater wireless sensor networks, is presented. The study make use of elements from $EL$-hyperstructures, ordered hyperstructures, or quasi-automata to show that the process of data aggregation could be interpreted as an hyperstructure generalization of an automaton.

The book ends with two articles on graph and hypergraph theory. A construction of granular structures using m-polar fuzzy hypergraphs and level hypergraphs is illustrated in Luqman et al. [21] by examples from a real-life problem. In Akram et al. [22], some properties related to edge regularity for q-rung picture fuzzy graphs are discussed. Besides, the paper presents the concept of ego-networks and its connections with q-rung picture fuzzy environment.

3. Conclusions

The publication of this Special Issue, the first one on hypercompositional algebra and its applications in an open access journal, has aroused the interest of many researchers in this field; therefore, a second Special Issue entitled “Hypercompositional Algebra and Applications” [23] was published, and several others are under development in specialized MDPI journals.