NeutroAlgebra

 

    From Paradoxism to Neutrosophy

Paradoxism is an international movement in science and culture, founded by Florentin Smarandache in 1980s, based on excessive use of antitheses, oxymoron, contradictions, and paradoxes. During three decades (1980-2020) hundreds of authors from tens of countries around the globe contributed papers to 15 international paradoxist anthologies.
In 1995, he extended the paradoxism (based on opposites) to a new branch of philosophy called neutrosophy (based on opposites and their neutral), that gave birth to many scientific branches, such as: neutrosophic logic, neutrosophic set, neutrosophic probability and statistics, neutrosophic algebraic structures, and so on with multiple applications in engineering, computer science, administrative work, medical research etc.
Neutrosophy is an extension of Yin-Yan Ancient Chinese Philosophy and of course of Dialectics.
 

    From Classical Algebraic Structures to NeutroAlgebraic Structures and AntiAlgebraic Structures

In 2019 and 2020 Smarandache [1, 2, 3] generalized the classical Algebraic Structures to NeutroAlgebraic Structures (or NeutroAlgebras) {whose operations and axioms are partially true, partially indeterminate, and partially false} as extensions of Partial Algebra, and to AntiAlgebraic Structures (or AntiAlgebras) {whose operations and axioms are totally false}.

The NeutroAlgebras & AntiAlgebras are a new field of research, which is inspired from our real world.

 

    Operation, NeutroOperation, AntiOperation

When we define an operation on a given set, it does not automatically mean that the operation is well-defined. There are three possibilities:
    1) The operation is well-defined (or inner-defined) for all set's elements (as in classical algebraic structures; this is classical Operation).
    2) The operation if well-defined for some elements, indeterminate for other elements, and outer-defined for others elements (this is NeutroOperation).
    3) The operation is outer-defined for all set's elements (this is AntiOperation).
 

    Axiom, NeutroAxiom, AntiAxiom

Similarly for an axiom, defined on a given set, endowed with some operation(s). When we define an axiom on a given set, it does not automatically mean that the axiom is true for all set’s elements. We have three possibilities again:
    1) The axiom is true for all set's elements [totally true] (as in classical algebraic structures; this is a classical Axiom).
    2) The axiom if true for some elements, indeterminate for other elements, and false for other elements (this is NeutroAxiom).
    3) The axiom is false for all set's elements (this is AntiAxiom).

 

    Algebra, NeutroAlgebra, AntiAlgebra

    1) An algebraic structure who’s all operations are well-defined and all axioms are totally true is called Classical Algebraic Structure (or Algebra).
    2) An algebraic structure that has at least one NeutroOperation or one NeutroAxiom (and no AntiOperation and no AntiAxiom) is called NeutroAlgebraic Structure (or NeutroAlgebra).
    3) An algebraic structure that has at least one AntiOperation or Anti Axiom is called AntiAlgebraic Structure (or AntiAlgebra).
    Therefore, a neutrosophic triplet structure is formed:
<Algebra, NeutroAlgebra, AntiAlgebra>.
“Algebra” can be: groupoid, semigroup, monoid, group, commutative group, ring, field, vector space, BCK-Algebra, BCI-Algebra, etc.
 


    Structure, NeutroStructure, AntiStructure

In general, Smarandache extended any classical Structure, in no matter what field of knowledge, to NeutroStructure and AntiStructure.
A classical Structure in any field of knowledge is composed of: some space, populated by some elements, and both (the space and the elements) been characterized by some relationships, hierarchies, laws, properties, ideas, shapes, etc.

    Similarly, we get the NeutroStructure and AntiStructure.

 

References

1. Florentin Smarandache: NeutroAlgebra is a Generalization of Partial Algebra. International Journal of Neutrosophic Science (IJNS), Volume 2, 2020, pp. 8-17. DOI: http://doi.org/10.5281/zenodo.3989285
http://fs.unm.edu/NeutroAlgebra.pdf

2. F. Smarandache, Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures, in Advances of Standard and Nonstandard Neutrosophic Theories, Pons Publishing House Brussels, Belgium, Chapter 6, pages 240-265, 2019; http://fs.unm.edu/AdvancesOfStandardAndNonstandard.pdf

3. Florentin Smarandache: Introduction to NeutroAlgebraic Structures and AntiAlgebraic Structures (revisited). Neutrosophic Sets and Systems, vol. 31, pp. 1-16, 2020. DOI: 10.5281/zenodo.3638232
http://fs.unm.edu/NSS/NeutroAlgebraic-AntiAlgebraic-Structures.pdf

4. Florentin Smarandache, Generalizations and Alternatives of Classical Algebraic Structures to NeutroAlgebraic Structures and AntiAlgebraic Structures, Journal of Fuzzy Extension and Applications (JFEA), J. Fuzzy. Ext. Appl. Vol. 1, No. 2 (2020) 85–87, DOI: 10.22105/jfea.2020.248816.1008
http://fs.unm.edu/NeutroAlgebra-general.pdf

5. A.A.A. Agboola, M.A. Ibrahim, E.O. Adeleke: Elementary Examination of NeutroAlgebras and AntiAlgebras viz-a-viz the Classical Number Systems. International Journal of Neutrosophic Science (IJNS), Volume 4, 2020, pp. 16-19. DOI: http://doi.org/10.5281/zenodo.3989530
http://fs.unm.edu/ElementaryExaminationOfNeutroAlgebra.pdf 

6. Agboola, A.A.A: Introduction to NeutroGroups. International Journal of Neutrosophic Science (IJNS), Volume 6, 2020, pp. 41-47. DOI: http://doi.org/10.5281/zenodo.3989823
http://fs.unm.edu/IntroductionToNeutroGroups.pdf

7. Agboola A.A.A: Introduction to NeutroRings. International Journal of Neutrosophic Science (IJNS), Volume 7, 2020, pp. 62-73. DOI: http://doi.org/10.5281/zenodo.3991389
http://fs.unm.edu/IntroductionToNeutroRings.pdf

8. Akbar Rezaei, Florentin Smarandache: On Neutro-BE-algebras and Anti-BE-algebras. International Journal of Neutrosophic Science (IJNS), Volume 4, 2020, pp. 8-15. DOI: http://doi.org/10.5281/zenodo.3989550
http://fs.unm.edu/OnNeutroBEalgebras.pdf

9. Mohammad Hamidi, Florentin Smarandache: Neutro-BCK-Algebra. International Journal of Neutrosophic Science (IJNS), Volume 8, 2020, pp. 110-117. DOI: http://doi.org/10.5281/zenodo.3991437
http://fs.unm.edu/Neutro-BCK-Algebra.pdf

10. Florentin Smarandache, Akbar Rezaei, Hee Sik Kim: A New Trend to Extensions of CI-algebras. International Journal of Neutrosophic Science (IJNS) Vol. 5, No. 1 , pp. 8-15, 2020; DOI: 10.5281/zenodo.3788124
http://fs.unm.edu/Neutro-CI-Algebras.pdf 

11. Florentin Smarandache: Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n-SuperHyperGraph, and Extension of HyperAlgebra to n-ary (Classical-/Neutro-/Anti-)HyperAlgebra. Neutrosophic Sets and Systems, Vol. 33, pp. 290-296, 2020. DOI: 10.5281/zenodo.3783103
http://fs.unm.edu/NSS/n-SuperHyperGraph-n-HyperAlgebra.pdf

 

 

 

 

MATHEMATICS Algebra Geometries Multispace
  Neutrosophic Environment Number Theory Statistics
  Plithogenic Set / Logic /      Probability / Statistics    
MATHEMATICS Algebra Geometries Multispace
  Neutrosophic Environment Number Theory Statistics
PHILOSOPHY Neutrosophy, a new branch of philosophy Law of Included Multiple-Middle & Principle of Dynamic Neutrosophic Opposition  
PHYSICS Absolute Theory of Relativity Quantum Paradoxes Unmatter
  Neutrosophic Physics Superluminal and Instantaneous Physics  
BIOLOGY Neutrosophic Theory of Evolution Syndrome  
ECONOMICS Poly-Emporium Theory    
LINGUISTICS Linguistic Paradoxes Linguistic Tautologies  
PSYCHOLOGY Neutropsychic Personality Illusion Law on Sensations and Stimuli
  Synonymity Test Complex  
SOCIOLOGY Social Paradox Sociological Theory  
LITERATURE pArAdOXisM oUTER-aRT Theatre
  Literatura Romana Bancuri  
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