Hypercompositional Algebra and Applications

Dear Colleagues,

This Special Issue is about Hypercompositional Algebra, which is a recent branch of abstract algebra.

As it is known, algebra is a generalization of arithmetic, where arithmetic (from the Greek word «ἀριθμός»—arithmós—meaning «number») is the area of mathematics that deals with numbers. During the classical period, Greek mathematicians created a geometric algebra where terms were represented by sides of geometric objects, while the Alexandrian School that followed (which was founded during the Hellenistic era) changed this approach dramatically. Especially, Diophantus made the first fundamental step towards symbolic algebra, as he developed a mathematical notation in order to write and solve algebraic equations.

In the Middle Ages that followed, the stage of mathematics shifted from the Greek world to the Arabic, and in AD 830 the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī wrote his famous treatise al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa’l-muqābala. After the 15th century AD, the stage of mathematics shifted to Europe, as the Islamic world was in decline and the European world was ascending. However, algebra remained essentially arithmetic with non-numerical mathematical objects until the 19th century, when the algebraic mathematical thought was changed radically because of the work of two young mathematicians: the Norwegian Niels Henrik Abel (1802–1829) and the Frenchman Evariste Galois (1811–1832), in algebraic equations. It became understood that the same processes could be applied to various objects or sets of entities other than numbers. So, abstract algebra was born.

The early years of the next (20th) century brought the end of determinism and certainty to science. This uncertainty affected algebra as well, via the work of a young French mathematician, Frederic Marty (1911–1940), who introduced an algebraic structure in which the rule of synthesizing elements gives a set of elements instead of one element only. He called this structure ‘’hypergroup’’, and he presented it during the 8th congress of Scandinavian Mathematicians, held in Stockholm in 1934. Unfortunately, Marty was killed at the age of 29, when his airplane was hit over the Baltic Sea while he was in military duty during World War II. His mathematical heritage on hypergroups is three papers only. However, other mathematicians such as H. Wall, M. Dresher, O. Ore, M. Krasner, and M. Kuntzmann started working on hypergroups shortly thereafter. Thus, hypercompositional algebra came into being as a branch of abstract algebra that deals with structures endowed with multi-valued operations. Multi-valued operations, also called ‘’hyperoperations’’ or ‘’hypercompositions’’, are laws of synthesis of the elements of a nonempty set, which associates a set of elements, instead of a single element, with every pair of elements.

Nowadays, this theory is characterized by huge diversity of character and content, and can present results in mathematics and other sciences such as physics, chemistry, biology, computer science, information technologies, social sciences, etc.

The aim of this Special Issue of the Journal Mathematics is dual: (a) to collect original research papers with new ideas and results on the contemporary research areas of hypercompositional algebra, as well as its applications to mathematics and other sciences and (b) to collect well-organized reviews on the aforementioned topics.

Prof. Dr. Christos G. Massouros
Guest Editor

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• Semihypergroup
• Quasihypergroup
• Hypergroup
• Hyperring
• Hyperfield
• Hypermodule
• Fuzzy hypercompositional structures
• Convex sets
• Graphs
• Formal languages
• Automata