2016-Sustainable Industrial Processing Summit
SIPS 2016 Volume 4: Santilli Intl. Symp. / Mathematics Applications
| Editors: | Kongoli F, Gaines G, Georgiev S, Bhalekar A |
| Publisher: | Flogen Star OUTREACH |
| Publication Year: | 2016 |
| Pages: | 320 pages |
| ISBN: | 978-1-987820-42-3 |
| ISSN: | 2291-1227 (Metals and Materials Processing in a Clean Environment Series) |
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Hypernumbers, Finite Hyper-Fields
Thomas Vougiouklis1;1DEMOCRITUS UNIVERSITY OF THRACE, Xanthi, Greece;
Type of Paper: Regular
Id Paper: 442
Topic: 38
Abstract:
Last decades hyperstructures have applications in mathematics and in other sciences. These applications range from biomathematics -conchology, inheritance- and hadronic physics or on leptons, in the Santilli’s iso-theory, to mention but a few. The hyperstructure theory is closely related to fuzzy theory; consequently, can be widely applicable in linguistic, in sociology, in industry and production, too. For these applications the largest class of the hyperstructures, the class Hv-structures, is used. The Hv-structures introduced in 1990 [4], satisfy the weak axioms where the non-empty intersection replaces the equality.
Algebraic hyperstructure is called a set H equipped with at least one hyperoperation (), abbreviate by hope: :HHP(H)-{}. The hyperstructure (H,) is called Hv-semigroup if it is weak associative: (xy)zx(yz), x,y,zH. It is called Hv-group if it is reproductive Hv-semigroup. It is called Hv-commutative group if, moreover, the weak commutativity: xyyx, x,yH, is valid.
The main tools of this theory are the fundamental relations which connect the Hv-structures with the corresponding classical ones by quotients. These relations are used to define hyperstructures as Hv-fields, Hv-vector spaces and so on, as well. The definition of the general hyperfield was not possible without the Hv-structures and their fundamental relations.
Hypernumbers or Hv-numbers are called the elements of Hv-fields. We present here this theory and focus on finite Hv-fields.