NeutroAlgebra & AntiAlgebra are generalizations of classical Algebras

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 Dialectics that have derived from the Yin-Yan Ancient Chinese Philosophy.

From Classical Algebraic Structures to NeutroAlgebraic Structures and AntiAlgebraic Structures

In 2019 Smarandache [1] 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} and on 2020 he continued to develop them [2,3,4].

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

In classical algebraic structures, all operations are 100% well-defined, and all axioms are 100% true,
but in real life, in many cases these restrictions are too harsh, since in our world we have things that only partially verify some operations or some laws.

Using the process of NeutroSophication of a classical algebraic structure we produce a NeutroAlgebra, while the process of AntiSophication of a classical algebraic structure produces an AntiAlgebra.

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 (also called inner-defined) for all set's elements [degree of truth T = 1] (as in classical algebraic structures; this is a classical Operation). Neutrosophically we write: Operation(1,0,0).
2) The operation if well-defined for some elements [degree of truth T], indeterminate for other elements [degree of indeterminacy I], and outer-defined for the other elements [degree of falsehood F], where (T,I,F) is different from (1,0,0) and from (0,0,1) (this is a NeutroOperation). Neutrosophically we write: NeutroOperation(T,I,F).
3) The operation is outer-defined for all set's elements [degree of falsehood F = 1] (this is an AntiOperation). Neutrosophically we write: AntiOperation(0,0,1).

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) [degree of truth T = 1] (as in classical algebraic structures; this is a classical Axiom). Neutrosophically we write: Axiom(1,0,0).
2) The axiom if true for some elements [degree of truth T], indeterminate for other elements [degree of indeterminacy I], and false for other elements [degree of falsehood F], where (T,I,F) is different from (1,0,0) and from (0,0,1) (this is NeutroAxiom). Neutrosophically we write NeutroAxiom(T,I,F).
3) The axiom is false for all set's elements [degree of falsehood F = 1](this is AntiAxiom). Neutrosophically we write AntiAxiom(0,0,1).

Theorem, NeutroTheorem, AntiTheorem

In any science, a classical Theorem, defined on a given space, is a statement that is 100% true (i.e. true for all elements of the space). To prove that a classical theorem is false, it is sufficient to get a single counter-example where the statement is false. Therefore, the classical sciences do not leave room for partial truth of a theorem (or a statement). But, in our world and in our everyday life, we have many more examples of statements that are only partially true, than statements that are totally true. The NeutroTheorem and AntiTheorem are generalizations and alternatives of the classical Theorem in any science.

Let's consider a theorem, stated on a given set, endowed with some operation(s). When we construct the theorem on a given set, it does not automatically mean that the theorem is true for all set’s elements. We have three possibilities again:
1) The theorem is true for all set's elements [totally true] (as in classical algebraic structures; this is a classical Theorem). Neutrosophically we write: Theorem(1,0,0).
2) The theorem if true for some elements [degree of truth T], indeterminate for other elements [degree of indeterminacy I], and false for the other elements [degree of falsehood F], where (T,I,F) is different from (1,0,0) and from (0,0,1) (this is a NeutroTheorem). Neutrosophically we write: NeutroTheorem(T,I,F).
3) The theorem is false for all set's elements (this is an AntiTheorem). Neutrosophically we write: AntiTheorem(0,0,1).

And similarly for (Lemma, NeutroLemma, AntiLemma), (Consequence, NeutroConsequence, AntiConsequence), (Algorithm, NeutroAlgorithm, AntiAlgorithm), (Property, NeutroProperty, AntiProperty), etc.

Algebra, NeutroAlgebra, AntiAlgebra

1) An algebraic structure who’s all operations are well-defined and all axioms are totally true is called a 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 a NeutroAlgebraic Structure (or NeutroAlgebra).
3) An algebraic structure that has at least one AntiOperation or one Anti Axiom is called an AntiAlgebraic Structure (or AntiAlgebra).
Therefore, a neutrosophic triplet is formed: <Algebra, NeutroAlgebra, AntiAlgebra>,
where “Algebra” can be any classical algebraic structure, such as: a groupoid, semigroup, monoid, group, commutative group, ring, field, vector space, BCK-Algebra, BCI-Algebra, etc.

Structure, NeutroStructure, AntiStructure in any field of knowledge

In general, by NeutroSophication, Smarandache extended any classical Structure, in no matter what field of knowledge, to a NeutroStructure, and by  AntiSophication to an AntiStructure.
A classical Structure, in any field of knowledge, is composed of: a non-empty space, populated by some elements, and both (the space and all elements) are characterized by some relations among themselves (such as: operations, laws, axioms, properties, functions, theorems, lemmas, consequences, algorithms, charts, hierarchies, equations, inequalities, etc.), and by their attributes (size, weight, color, shape, location, etc.).

Of course, when analysing a structure, it counts with respect to what relations and what attributes we do it.

Relation, NeutroRelation, AntiRelation

1) A classical Relation is a relation that is true for all elements of the set (degree of truth T = 1). Neutrosophically we write Relation(1,0,0).

2) A NeutroRelation is a relation that is true for some of the elements (degree of truth T), indeterminate for other elements (degree of indeterminacy I), and false for the other elements (degree of falsehood F). Neutrosophically we write Relation(T,I,F), where (T,I,F) is different from (1,0,0) and (0,0,1).

3) An AntiRelation is a relation that is false for all elements (degree of falsehood F = 1). Neutrosophically we write Relation(0,0,1).

Attribute, NeutroAttribute, AntiAttribute

1) A classical Attribute is an attribute that is true for all elements of the set (degree of truth T = 1). Neutrosophically we write Attribute(1,0,0).

2) A NeutroAttribute is an attribute that is true for some of the elements (degree of truth T), indeterminate for other elements (degree of indeterminacy I), and false for the other elements (degree of falsehood F). Neutrosophically we write Attribute(T,I,F), where (T,I,F) is different from (1,0,0) and (0,0,1).

3) An AntiAttribute is an attribute that is false for all elements (degree of falsehood F = 1). Neutrosophically we write Attribute(0,0,1).

Structure, NeutroStructure, AntiStructure

1) A classical Structure is a structure whose all elements are characterized by the same given Relationships and Attributes and Laws.

2) A NeutroStructure is a structure that has at least one NeutroRelation or one NeutroAttribute or one NeutroLaw, and no AntiRelation and no AntiAttribute and no AntiLaw.

3) An AntiStructure is a structure that has at least one AntiRelation or one AntiAttribute or one AntiLaw.

Almost all Structures are NeutroStructures

The Classical Structures in science mostly exist in theoretical, abstract, perfect, homogeneous, idealistic spaces - because in our everyday life almost all structures are NeutroStructures, since they are neither perfect nor applying to the whole population, and not all elements of the space have the same relations and same attributes in the same degree (not all elements behave in the same way).

The indeterminacy and partiality, with respect to the space, to their elements, to their relations or to their attributes are not taken into consideration in the Classical Structures. But our Real World is full of structures with indeterminate (vague, unclear, conflicting, unknown, etc.) data and partialities.

There are exceptions to almost all laws, and the laws are perceived in different degrees by different people.

Examples of NeutroStructures from our Real World

(i) In the Christian society the marriage law is defined as the union between a male and a female (degree of truth).

But, in the last decades, this law has become less than 100% true, since persons of the same sex were allowed to marry as well (degree of falsehood).

On the other hand, there are transgender people (whose sex is indeterminate), and people who have changed the sex by surgical procedures, and these people (and their marriage) cannot be included in the first two categories (degree of indeterminacy).

Therefore, since we have a NeutroLaw (with respect to the Law of Marriage) we have a Christian NeutroStructure.

(ii) In India, the law of marriage is not the same for all citizen: Hindi religious men may marry only one wife, while the Muslims may marry up to four wives.

(iii) Not always the difference between good and bad may be clear, from a point of view a thing may be good, while from another point of view bad. There are things that are partially good, partially neutral, and partially bad.

(iv) The laws do not equally apply to all citizens, so they are NeutroLaws. Some laws apply to some degree to a category of citizens, and to a different degree to another category. Almost always there are exceptions to the law! As such, there is an American folkloric joke: All people are born equal, but some people are more equal than others!

- There are powerful people that are above the laws, and other people that benefit of immunity with respect to the laws.

- For example, in the court of law, privileged people benefit from better defense lawyers than the lower classes, so they may get a lighter sentence.

- Not all criminals go to jail, but only those caught and proven guilty in the court of law. Nor criminals that for reason of insanity cannot stand trail and do not go to jail since they cannot make a difference between right and wrong.

- Unfortunately, even innocent people went and might go to jail because of sometimes jurisdiction mistakes...

- The Hypocrisy and Double Standard are widely spread: some regulation applies to some people, but not to others!

(v) Anti-Abortion Law does not apply to all pregnant women: the incest, rapes, and women whose life is threatened may get abortions.

(vi) Gun-Control Law does not apply to all citizen: the police, army, security, professional hunters are allowed to bear arms.

Etc.

References

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

2. 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

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. A.A.A. Agboola: 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. A.A.A. Agboola: 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

12. A.A.A. Agboola: On Finite NeutroGroups of Type-NG. International Journal of Neutrosophic Science (IJNS), Volume 10, Issue 2, 2020, pp. 84-95. DOI: 10.5281/zenodo.4277243, http://fs.unm.edu/IJNS/OnFiniteNeutroGroupsOfType-NG.pdf

13. A.A.A. Agboola: On Finite and Infinite NeutroRings of Type-NR. International Journal of Neutrosophic Science (IJNS), Volume 11, Issue 2, 2020, pp. 87-99. DOI: 10.5281/zenodo.4276366, http://fs.unm.edu/IJNS/OnFiniteAndInfiniteNeutroRings.pdf

14. A.A.A. Agboola, Introduction to AntiGroups, International Journal of Neutrosophic Science (IJNS), Vol. 12, No. 2, PP. 71-80, 2020, http://fs.unm.edu/IJNS/IntroductionAntiGroups.pdf

15. M.A. Ibrahim and A.A.A. Agboola, Introduction to NeutroHyperGroups, Neutrosophic Sets and Systems, vol. 38, 2020, pp. 15-32. DOI: 10.5281/zenodo.4300363,

16. Elahe Mohammadzadeh and Akbar Rezaei, On NeutroNilpotentGroups, Neutrosophic Sets and Systems, vol. 38, 2020, pp. 33-40. DOI: 10.5281/zenodo.4300370, http://fs.unm.edu/NSS/OnNeutroNilpotentGroups3.pdf

17. F. Smarandache, Structure, NeutroStructure, and AntiStructure in Science, International Journal of Neutrosophic Science (IJNS), Volume 13, Issue 1, PP: 28-33, 2020; http://fs.unm.edu/IJNS/NeutroStructure.pdf

18. Diego Silva Jiménez, Juan Alexis Valenzuela Mayorga, Mara Esther Roja Ubilla, and Noel Batista HernándezNeutroAlgebra for the evaluation of barriers to migrants’ access in Primary Health Care in Chile based on PROSPECTOR functionNeutrosophic Sets and Systems, vol. 39, 2021, pp. 1-9. DOI: 10.5281/zenodo.4444189

19. Madeleine Al-Tahan, F. Smarandache, and Bijan DavvazNeutroOrderedAlgebra: Applications to SemigroupsNeutrosophic Sets and Systems, vol. 39, 2021, pp.133-147. DOI: 10.5281/zenodo.4444331

20. F. Smarandache, Universal NeutroAlgebra and Universal AntiAlgebra, Chapter 1, pp. 11-15, in the collective book NeutroAlgebra Theory, Vol. 1, edited by F. Smarandache, M. Sahin, D. Bakbak, V. Ulucay, A. Kargin, Educational Publ., Grandview Heights, OH, United States, 2021

21. Madeleine Al-Tahan, NeutroOrderedAlgebra: Theory and Examples, 3rd International Workshop on Advanced Topics in Dynamical Systems, University of Kufa, Iraq, March 1st, 2021.

22. F. Smarandache A. Rezaei A.A.A. Agboola Y.B. Jun R.A. Borzooei B. Davvaz A. Broumand Saeid M. Akram M. Hamidi S. Mirvakilii, On NeutroQuadrupleGroups, 51st Annual Mathematics Conference, Kashan, February 16-19, 2021.

23. Madeleine Al-Tahan, Bijan Davvaz, Florentin Smarandache, and Osman Anis, On Some NeutroHyperstructures, Symmetry 2021, 13, 535, pp. 1-12, https://doi.org/10.3390/sym13040535; http://fs.unm.edu/NeutroHyperstructure.pdf

24. A. Rezaei, F. Smarandache, and S. Mirvakili, Applications of (Neutro/Anti)sophications to Semihypergroups, Journal of Mathematics, Hindawi, vol. 2021, Article ID 6649349, pp. 1-7, 2021; https://doi.org/10.1155/2021/6649349.

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