Hyperstructures in Lie-Santilli Admissibility and Iso-Theories

Maria Santilli Ruggero, Thomas Vougiouklis

Abstract


In the quiver of hyperstructures Professor R. M. Santilli, in early 90'es, tried to find algebraic structures in order to express his pioneer Lie-Santilli's Theory. Santilli's theory on 'isotopies' and 'genotopies', born in 1960's, desperately needs 'units e' on left or right, which are nowhere singular, symmetric, real-valued, positive-defined for n-dimensional matrices based on the so called isofields.These elements can be found in hyperstructure theory, especially in $H_v$-structure theory introduced in 1990. This connection appeared first in 1996 and actually several $H_v$-fields, the e-hyperfields, can be used as isofields or genofields so as, in such way they should cover additional properties and satisfy more restrictions. Several large classes of hyperstructures as the P-hyperfields, can be used in Lie-Santilli's theory when multivalued problems appeared, either in finite or in infinite case. We review some of these topics and we present the Lie-Santilli admissibility in Hyperstructures.

Keywords


Lie-Santilli iso-theory; hyperstructures; hope; $H_v$-structures

Full Text:

PDF

References


R. Anderson, A. A. Bhalekar, B. Davvaz, P. S. Muktibodh, T. Vougiouk-lis, An introduction to Santilli’s isodual theory of antimatter and the open problem of detecting antimatter asteroids, NUMTA B., 6 (2012-13), 1–33.

P. Corsini, V. Leoreanu, Application of Hyperstructure Theory, Klower Ac. Publ.:(2003).

P. Corsini, T. Vougiouklis, From groupoids to groups through hypergroups, Rend. Mat. VII, 9, (1989), 173–181.

B. Davvaz, V. Leoreanu-Fotea, Hyperring Theory and Applications, Int. Acad. Press, USA: (2007).

B. Davvaz, R.M. Santilli, T. Vougiouklis, Multi-valued Hypermathematics for characterization of matter and antimatter systems, J. Comp. Meth. Sci. Eng. (JCMSE) 13, (2013), 37–50.

B. Davvaz, R.M. Santilli, T. Vougiouklis, Algebra, Hyperalgebra and Lie-Santilli Theory, J. Generalized Lie Theory Appl., (2015), 9:2, 1–5.

A. Dramalidis, T. Vougiouklis, Lie-Santilli Admissibility on non-square ma-trices with the helix hope, CACAA, 4, N. 4, (2015), 353–360.

S. Georgiev, Foundations of Iso-Differential Carlculus, Nova Sc. Publ., V.1-6: (2016).

P. Nikolaidou, T. Vougiouklis, The Lie-Santilli admissible hyperalgebras of type An, Ratio Math. 26, (2014), 113–128.

R. M. Santilli, Embedding of Lie-algebras into Lie-admissible algebras, Nuovo Cimento 51, 570: (1967).

R. M. Santilli, An introduction to Lie-admissible algebras, Suppl. Nuovo Cimento, 6, 1225: (1968).

R. M. Santilli, Dissipativity and Lie-admissible algebras, Mecc.1, 3: (l969).

R. M. Santilli, Representation of antiparticles via isodual numbers, spaces and geometries, Comm. Theor. Phys. 3, (1994), 153–181

R. M. Santilli, Hadronic Mathematics, Mechanics and Chemistry, Volumes I, II, III, IV and V, International Academic Press, USA: (2007).

R.M. Santilli, T. Vougiouklis, Isotopies, Genotopies, Hyperstructures and their Applications, New frontiers in Hyperstructures, Hadronic, (1996), 1– 48.

R. M. Santilli, T. Vougiouklis, Lie-admissible hyperalgebras, Italian J. Pure Appl. Math., N.31, (2013), 239–254.

T. Vougiouklis, The fundamental relation in hyperrings. The general hyper-field, 4th AHA, Xanthi 1990, World Scientific, (1991), 203–211

T. Vougiouklis, Hyperstructures and their representations, Hadronic Press Inc.: (1994).

T. Vougiouklis, Some remarks on hyperstructures, Contemporary Math., Amer. Math. Society, 184, (1995), 427–431.

T. Vougiouklis, On Hv-rings and Hv-representations, Discrete Math., Else-vier, 208/209, (1999), 615–620.

T. Vougiouklis, Hyperstructures in isotopies and genotopies, Advances in Equations and Inequalities, Hadronic Press, (1999), 275–291.

T. Vougiouklis, A hyperoperation defined on a groupoid equipped with a map, Ratio Mat., N.1, (2005), 25–36.

T. Vougiouklis, @-operations and Hv-fields, Acta Math. Sinica, English S., V.23, 6, (2008), 965–972.

T. Vougiouklis, The Santilli’s theory ’invasion’ in hyperstructures, AGG, 28(1), (2011), 83–103.

T. Vougiouklis, The e-hyperstructures, J. Mahani Math. Research Center, V.1, N.1, (2012), 13–28.

T. Vougiouklis, The Lie-hyperalgebras and their fundamental relations, Southeast Asian Bull. Math., V.37(4), (2013), 601–614.

T. Vougiouklis, On the isoHv-numbers, Hadronic J., Dec.5, (2014), 1–18.

T. Vougiouklis, Lie-Santilli Admissibility using P-hyperoperations on matri-ces, Hadronic J., Dec.7, (2014), 1–14.

T. Vougiouklis, Iso-hypernumbers, Iso-Hv-numbers, ICNAAM 2014, AIP 1648, 510019, (2015); http://dx.doi.org/10.1063/1.4912724

T. Vougiouklis, Lie-Santilli Admissibility on non square matrices, Proc. IC-NAAM 2014, AIP 1648, (2015);http://dx.doi.org/10.1063/1.4912725

T. Vougiouklis, Hypermathematics, Hv-structures, hypernumbers, hyper-matrices and Lie-Santilli admissibility, American J. Modern Physics, 4(5), (2015), 34–46.

T. Vougiouklis, Iso-Hv-numbers, Clifford Analysis, Clifford Alg. Appl. CA-CAA, V. 4, N. 4, (2015), 345–352.

T. Vougiouklis, S. Vougiouklis, The helix hyperoperations, Italian J. Pure Appl. Math., N.18, (2005), 197–206.

T. Vougiouklis, S. Vougiouklis, Hyper Lie-Santilli admisibility, AGG, 33, N.4, (2016), 427–442.




DOI: http://dx.doi.org/10.23755/rm.v33i0.374

Refbacks

  • There are currently no refbacks.


Copyright (c) 2017 Maria Santilli Ruggero, Thomas Vougiouklis

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.