From structures to weak hyperstructures

Document Type : Research Paper

Author

School of Science of Education, Democritus University of Thrace, 68100 Alexandroupolis, Greece

Abstract

The hyperstructures, born in 1934, are a generalization of the operation by the hyper-operation or multivalued operation. In 1970, the $\beta^*$ relation, called now fundamental, connects structures with corresponding hyperstructures. In 1990, a generalization on the properties or axioms, which the hyper-operations fulfill, called weak properties, appeared. So the $H_v$-structures were born. The study of hyperstructures, especially of the weak ones, gives new topics in the field which have numerous of applications in mathematics and other sciences. Here, we present an overview of applications of $H_v$-structures.

Keywords

Main Subjects

References

[1] R. Borzooei, A.B. Saeid, R. Ameri, A. Rezaei, Involutory BE-algebras, J. Math. Appl. 37, 2014, 13-26.
[2] P. Corsini, V. Leoreanu, Application of Hyperstructure Theory, Kluwer Acad. Publ., 2003.
[3] P. Corsini, T. Vougiouklis, From groupoids to groups through hypergroups, Rend. Mat. VII, 9, 1989, 173-181.
[4] B. Davvaz, Semihypergroup Theory, Academic Press, Elsevier, 2016.
[5] B. Davvaz, A brief survey on algebraic hyperstructures: Theory and applications, JAHLA, V1, N3, 2020, 15-29.
[6] B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, Int. Acad. Press, USA, 2007.
[7] B. Davvaz, V. Leoreanu-Fotea, T. Vougiouklis, A survey on the Theory of n-Hypergroups, Reviews Math. & Appl., 2023, 1-17.
[8] B. Davvaz, R.M. Santilli, T. Vougiouklis, Algebra, Hyperalgebra and Lie-Santilli Theory, J. Gen. Lie Th. Appl., 9:2, 2015, 1-5.
[9] B. Davvaz, S. Vougioukli, T. Vougiouklis, On the multiplicative Hv-rings derived from helix hyperoperations, Util. Math., 84, 2011, 53-63.
[10] B. Davvaz, T. Vougiouklis, A Walk Through Weak Hyperstructures, Hv-Structures, World Scienti c, 2018.
[11] S. Ostadhadi-Dehkordi, T. Vougiouklis, K. Hila, H-sets and applications on Hv-groups, J. Algebraic Systems, 10(1), 2022, 79-93.
[12] R.M. Santilli, Hadronic Mathematics, Mechanics and Chemistry, Vol. I, II, III, IV, V, Int. Academic Press, USA, 2007.
[13] R.M. Santilli, T. Vougiouklis, Isotopies, Genotopies, Hyperstructures and their Applications, New frontiers Hyperstr., Hadronic, 1996, 1-48.
[14] R.M. Santilli, T. Vougiouklis, Lie-admissible hyperalgebras, Italian J. Pure Appl. Math., 31, 2013, 239-254.
[15] S. Vougiouklis, Hv-vector spaces from helix hyperoperations, Int. J. Math. Anal. NS, 1(2), 2009, 109-120.
[16] S. Vougioukli, Helix hyperoperation in teaching research, Science & Philosophy, 8(2), 2020, 157-163.
[17] T. Vougiouklis, Generalization of P-hypergroups, Rend. Circolo Mat. Palermo Ser. II, 36, 1987, 114-121.
[18] T. Vougiouklis, Groups in hypergroups, Annals Discrete Math. 37, 1988, 459-468.
[19] T. Vougiouklis, The very thin hypergroups and the S-construction, Combinatorics '88, Incidence Geom. Comb. Str., 2, 1991, 471-477.
[20] T. Vougiouklis, The fundamental relation in hyperrings. The general hyper eld, 4thAHA, Xanthi 1990, World Scienti c, 1991, 203-211.
[21] T. Vougiouklis, Representations of hypergroups by generalized permutations, Algebra Universalis, 29, 1992, 172-183.
[22] T. Vougiouklis, Hyperstructures and their Representations, Monographs in Math., Hadronic, 1994.
[23] T. Vougiouklis, Some remarks on hyperstructures, Contemporary Math., American Math. Society, 184, 1995, 427-431.
[24] T. Vougiouklis, Convolutions on WASS hyperstructures, Discrete Math., 174, 1997, 347-355.
[25] T. Vougiouklis, Enlarging Hv-structures, Algebras and Combinatorics, ICAC'97, Hong Kong, Springer{Verlag, 1999, 455-463.
[26] T. Vougiouklis, On Hv-rings and Hv-representations, Discrete Math., Elsevier, 208/209, 1999, 615-620.
[27] T. Vougiouklis, @-operations and Hv- elds, Acta Math. Sinica, English S., V23, 6, 2008, 965-972.
[28] T. Vougiouklis, The e-hyperstructures, J. Mahani Math. Research Center, V1, N1, 2012, 13-28.
[29] T. Vougiouklis, From Hv-rings to Hv- elds, Int. J. Alg. Hyperstr. Appl., V1, N1, 2014, 1-13.
[30] T. Vougiouklis, Minimal Hv- elds, Ratio Mathematica V38, 2020, 313-328.
[31] T. Vougiouklis, Fundamental Relations in Hv-structures. The `Judging from the Results' proof, JAHLA, V1, N1, 2020, 21-36.
[32] T. Vougiouklis, Philosophical suggestions by the single elements, Sc. & Philosophy V9(1), 2021, 45-53.
[33] T. Vougiouklis, Representations on Raised Very Thin Hv- elds, TFSS, V1, N1, 2022, 88-105.
[34] T. Vougiouklis, P. Kambaki-Vougioukli, On the use of the bar, China-USA Business Review, V10, N6, 2011, 484-489.
[35] T. Vougiouklis, S. Vougiouklis, The helix hyperoperations, Italian J. Pure Appl. Math. N18, 2005, 197-206.
[36] T. Vougiouklis, S. Vougiouklis, Helix-Hopes on Finite Hyper elds, Ratio Math., V31, 2016, 65-78.
[37] M.M. Zahedi, A.B. Saeid, R.A. Borzooei, Implicative hyper K-algebras, Czechoslovak Math. J., V55, 2005, 439-453.

Files

History

  • Receive Date: 18 February 2025
  • Revise Date: 08 May 2025
  • Accept Date: 01 July 2025

Share

How to cite

Statistics