
 NeutroAlgebra & AntiAlgebra are generalizations of classical Algebras
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 (19802020) hundreds of
authors from tens of countries around the globe contributed papers to 15
international paradoxist anthologies. 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% welldefined, 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 welldefined. There are three
possibilities: 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:
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 counterexample 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: 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 welldefined and all axioms are totally true is called a classical
Algebraic Structure (or Algebra). 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. 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) AntiAbortion Law does not apply to all pregnant women: the incest, rapes, and women whose life is threatened may get abortions. (vi) GunControl 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/NeutroAlgebraicStructureschapter.pdf ], in his book Advances of Standard and Nonstandard Neutrosophic Theories, Pons Publishing House Brussels, Belgium, Chapter 6, pages 240265, 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. 817. DOI:
http://doi.org/10.5281/zenodo.3989285 3. Florentin Smarandache: Introduction to NeutroAlgebraic
Structures and AntiAlgebraic Structures (revisited). Neutrosophic Sets and
Systems, vol. 31, pp. 116, 2020. DOI: 10.5281/zenodo.3638232 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
6. A.A.A. Agboola: Introduction to NeutroGroups.
International Journal of Neutrosophic Science (IJNS), Volume 6, 2020, pp. 4147.
DOI:
http://doi.org/10.5281/zenodo.3989823
7. A.A.A. Agboola: Introduction to NeutroRings. International
Journal of Neutrosophic Science (IJNS), Volume 7, 2020, pp. 6273. DOI:
http://doi.org/10.5281/zenodo.3991389 8. Akbar Rezaei, Florentin Smarandache: On NeutroBEalgebras
and AntiBEalgebras. International Journal of Neutrosophic Science (IJNS),
Volume 4, 2020, pp. 815. DOI:
http://doi.org/10.5281/zenodo.3989550 9. Mohammad Hamidi, Florentin Smarandache: NeutroBCKAlgebra.
International Journal of Neutrosophic Science (IJNS), Volume 8, 2020, pp.
110117. DOI:
http://doi.org/10.5281/zenodo.3991437 10. Florentin Smarandache, Akbar Rezaei, Hee Sik Kim: A New
Trend to Extensions of CIalgebras. International Journal of Neutrosophic
Science (IJNS) Vol. 5, No. 1 , pp. 815, 2020; DOI: 10.5281/zenodo.3788124 11. Florentin Smarandache: Extension of HyperGraph to nSuperHyperGraph
and to Plithogenic nSuperHyperGraph, and Extension of HyperAlgebra to nary
(Classical/Neutro/Anti)HyperAlgebra. Neutrosophic Sets and Systems, Vol. 33,
pp. 290296, 2020. DOI: 10.5281/zenodo.3783103
12. A.A.A. Agboola: On Finite NeutroGroups of TypeNG. International Journal
of Neutrosophic Science (IJNS), Volume 10, Issue 2, 2020, pp. 8495. DOI:
10.5281/zenodo.4277243,
http://fs.unm.edu/IJNS/OnFiniteNeutroGroupsOfTypeNG.pdf
14. A.A.A. Agboola, Introduction to AntiGroups, International Journal of Neutrosophic Science (IJNS), Vol. 12, No. 2, PP. 7180, 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. 1532. DOI: 10.5281/zenodo.4300363, http://fs.unm.edu/NSS/IntroductionToNeutroHyperGroups2.pdf 16. Elahe Mohammadzadeh and Akbar Rezaei, On NeutroNilpotentGroups, Neutrosophic Sets and Systems, vol. 38, 2020, pp. 3340. 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: 2833, 2020; http://fs.unm.edu/IJNS/NeutroStructure.pdf 18. Diego Silva Jiménez, Juan Alexis Valenzuela Mayorga, Mara Esther Roja Ubilla, and Noel Batista Hernández, NeutroAlgebra for the evaluation of barriers to migrants’ access in Primary Health Care in Chile based on PROSPECTOR function, Neutrosophic Sets and Systems, vol. 39, 2021, pp. 19. DOI: 10.5281/zenodo.4444189 19. Madeleine AlTahan, F. Smarandache, and Bijan Davvaz, NeutroOrderedAlgebra: Applications to Semigroups, Neutrosophic Sets and Systems, vol. 39, 2021, pp.133147. DOI: 10.5281/zenodo.4444331 20. F. Smarandache, Universal NeutroAlgebra and Universal AntiAlgebra, Chapter 1, pp. 1115, 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 AlTahan, 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 1619, 2021. 23. Madeleine AlTahan, Bijan Davvaz, Florentin Smarandache, and Osman Anis, On Some NeutroHyperstructures, Symmetry 2021, 13, 535, pp. 112, 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. 17, 2021; https://doi.org/10.1155/2021/6649349. 25. F. Smarandache, M. AlTahan (editors), Theory and Applications of NeutroAlgebras as Generalizations of Classical Algebras, IGI Global, USA, 2022, new


