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
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-Yang Ancient Chinese Philosophy.
In 2019 Smarandache [1] generalized the classical
Algebraic Structures to NeutroAlgebraic Structures
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].
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.
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
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/NA/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/NA/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/NA/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/NA/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,
http://fs.unm.edu/NSS/IntroductionToNeutroHyperGroups2.pdf
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/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. 1-9. DOI: 10.5281/zenodo.4444189
19. Madeleine Al-Tahan,
F. Smarandache, and Bijan Davvaz, NeutroOrderedAlgebra:
Applications to Semigroups, Neutrosophic
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.
25. F.
Smarandache, M. AlTahan (editors), Theory
and Applications of NeutroAlgebras as
Generalizations of Classical
Algebras,
IGI Global, USA, 2022, new