

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.
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
YinYang 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%
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:
1) The operation is welldefined (also called innerdefined) 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 welldefined for some elements [degree of truth
T], indeterminate for other elements [degree of indeterminacy I], and
outerdefined 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 outerdefined 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 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:
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
welldefined 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, BCKAlgebra, BCIAlgebra, etc.
Topology, NeutroTopology, AntiTopology
1) A topology who’s all axioms are totally true is called a classical
Topology (or Topology).
2) A topology that has at least one NeutroAxiom (and no AntiAxiom) is called a
NeutroTopology.
3) A topology that has at least one AntiAxiom is
called an AntiTopology.
Therefore, a neutrosophic triplet is formed: <Topology,
NeutroTopology,
AntiTopology>,
where “Topology” may be any type of classical topology.
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
nonempty 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) 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
http://fs.unm.edu/NeutroAlgebra.pdf
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
http://fs.unm.edu/NSS/NeutroAlgebraicAntiAlgebraicStructures.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/NeutroAlgebrageneral.pdf
5. A.A.A. Agboola, M.A. Ibrahim, E.O. Adeleke: Elementary Examination of
NeutroAlgebras and AntiAlgebras vizaviz the Classical Number Systems.
International Journal of Neutrosophic Science (IJNS), Volume 4, 2020,
pp. 1619. 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. 4147. 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. 6273. DOI: http://doi.org/10.5281/zenodo.3991389
http://fs.unm.edu/IntroductionToNeutroRings.pdf
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
http://fs.unm.edu/NA/OnNeutroBEalgebras.pdf
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
http://fs.unm.edu/NeutroBCKAlgebra.pdf
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
http://fs.unm.edu/NeutroCIAlgebras.pdf
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
http://fs.unm.edu/NSS/nSuperHyperGraphnHyperAlgebra.pdf
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/NA/OnFiniteNeutroGroupsOfTypeNG.pdf
13. A.A.A. Agboola: On Finite and Infinite NeutroRings of TypeNR.
International Journal of Neutrosophic Science (IJNS), Volume 11, Issue
2, 2020, pp. 8799. DOI: 10.5281/zenodo.4276366, http://fs.unm.edu/NA/OnFiniteAndInfiniteNeutroRings.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/NA/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/NA/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 