Plithogeny,
Plithogenic Set / Logic / Probability / Statistics,
and Symbolic
Plithogenic Algebraic Structures
1. Etymology of Plithogeny
Plitho-geny etymologically comes from: (Gr.) πλήθος (plithos)
= crowd, large number of, multitude, plenty of, and -geny < (Gr.) -γενιά (-geniá)
= generation, the production of something & γένeια (géneia) = generations, the
production of something < -γένεση (-génesi) = genesis, origination, creation,
development, according to Translate Google Dictionaries [ https://translate.google.com/
] and Webster’s New World Dictionary of American English, Third College Edition,
Simon & Schuster, Inc., New York, pp. 562-563, 1988.
Therefore, plithogeny is
the genesis or origination, creation, formation, development, and evolution of
new entities from dynamics and organic fusions of contradictory and/or neutrals
and/or non-contradictory multiple old entities.
Plithogenic means what is pertaining to plithogeny.
All (plithogeny,
plithogenic set, plithogenic logic, plithogenic probability, plithogenic statistics, and
symbolic plithogenic algebraic structures) were introduced in 2018-2019 by F.
Smarandache.
2.1. Plithogenic
Set
A plithogenic set P is a set whose elements are
characterized by one or more attributes, and each attribute may have many
values. Each attribute’s value v has a corresponding degree of appurtenance
d(x,v) of the element x to the set P, with respect to some given criteria.
In order to obtain a better accuracy for the plithogenic
aggregation operators, a contradiction (dissimilarity) degree is defined between
each attribute value and the dominant (most important) attribute value.
{However, there are cases when such dominant attribute value may not be
taking into consideration or may not exist [therefore it is considered zero by
default], or there may be many dominant attribute values. In such cases, either
the contradiction degree function is suppressed, or another relationship
function between attribute values should be established.}
The plithogenic aggregation operators (intersection,
union, complement, inclusion, equality) are based on contradiction degrees
between attributes’ values, and the first two are linear combinations of the
fuzzy operators’ tnorm and tconorm.
Plithogenic set was introduced by Smarandache in 2017 [see
B1] and it is a generalization of the crisp set,
fuzzy set, intuitionistic fuzzy set, neutrosophic set, since these these types
of sets are characterized by a single attribute value (appurtenance): which has
one value (membership) – for the crisp set and fuzzy set, two values
(membership, and nonmembership) – for intuitionistic fuzzy set, or three values
(membership, nonmembership, and indeterminacy) – for neutrosophic set.
2.2. Example of Plithogenic Set
Let P be a plithogenic set, representing the students from a college. Let x
belonging to P
be a generic student that is characterized by three attributes:
- altitude (a), whose values are {tall, short} = {a1, a2};
- weight (w), whose values are {obese, fat, medium, thin} = {w1, w2, w3, w4};
and
- hair color, whose values are {blond, reddish, brown} = {h1, h2, h3}.
The multi-attribute of dimension 3 is
Let's say P = {John(a1, w3, h2), Richard(a1, w3, h2)} = {John(tall,
thin, reddish), Richard(tall, thin, reddish}.
From the view point of expert A, one has
PA= {John(0.7,
0.2, 0.4), Richard(0.5, 0.8, 0.6)},
which means that, from the
view point of expert A, John's fuzzy degrees of tallness,
thinness, and reddishness are respectively
0.7, 0.2, and 0.4; while Richard's fuzzy degrees of
tallness,
thinness, and reddishness are respectively
0.5, 0.8, and 0.6.
While from the view point of expert B,
one has PB = {John(0.8,
0.2, 0.5), Richard(0.3, 0.7, 0.4)}.
The uni-dimensional attribute contradiction degrees
are:
c(a1, a2) = 1;
c(w1, w2) = 1/3; c(w1,
w3) = 2/3; c(w1, w4) = 1;
c(h1, h2) = 1/2; c(h1,
h3) =1.
Dominant attribute values are:
,
and respectively
for each corresponding uni-dimensional attribute a, w, and h respectively. Let’s use the fuzzy
tnorm = x /\F y = xy {where F
means fuzzy conjunction}, and
fuzzy tconorm = x \/F y = F
y = x + y – xy {where F
means fuzzy disjunction}.
Then:
(x, y, z) /\N (d, e, f) = (xd, (1/2)[(ye) + (y+e-ye)], z+f-zf) = (xd, (y+e)/2,
z+f-zf).
2.3. A plithogenic application to images
A
pixel x may be characterized by colors c1, c2, ..., cn. We write x(c1, c2, ...,
cn), where n >= 1.
We
may consider the degree of each color either fuzzy, intuitionistic fuzzy, or
neutrosophic.
For example.
Fuzzy degree:
x(0.4, 0.6, 0.1, ..., 0.3).
Intuitionistic fuzzy degree:
x(
(0.1, 0.2), (0.3, 0.5), (0.0, 0.6), ..., (0.8, 0.9) )
Neutrosophic degree:
x(
(0.0, 0.3, 0.6), (0.2, 0.8, 0.9), (0.7, 0.4, 0.2), ..., (0.1, 0.1, 0.9) )
Then, we can use a plithogenic operator to combine them.
For example:
x(0.4, 0.6, 0.1, ..., 0.3) /\P x(0.1,
0.7, 0.5, ..., 0.2) =
...
We
first establish the degrees of contradictions between between all colors ci and
cj in
order to find the linear combinations of t-norm and
t-conorm that one applies to
each color
(similar to the indeterminacy above).
3.1. Plithogenic Logic
A plithogenic logic proposition P is a proposition that is characterized by
degrees of many truth-values with respect to the corresponding
attribute's value. For each attribute’s value v there is a
corresponding degree of truth-value d(P, v) of P with respect to the attribute
value v. Plithogenic logic is a generalization of the classical logic,
fuzzy logic, intuitionistic fuzzy logic, and neutrosophic logic, since these
four types of logics are characterized by a single attribute value
(truth-value): which has one value (truth) – for the classical logic and
fuzzy logic, two values (truth, and falsehood) – for intuitionistic
fuzzy logic, or three values (truth, falsehood, and indeterminacy) – for
neutrosophic logic. A plithogenic logic proposition P, in general, may
be characterized by more than four degrees of truth-values resulted
from under various attributes.
3.2. Example of Plithogenic Logic
Let P = “John is a knowledgeable person” be a logical proposition. It is
evaluated by multiple experts (two) and under multiple attribute values
(six), that's why it is called plithogenic.
The three attributes under which this proposition has to be evaluated
about - according to the experts - are: Science (whose attribute values
are: mathematics, physics, anatomy), Literature (whose attribute values are: poetry, novel), and Arts (whose only attribute value is:
sculpture).
According to Expert A(lexander), the truth-values of plithogenic proposition
P are: PA(0.7, 0.6, 0.4; 0.9, 0.2; 0.5),which means that John’s
degree of truth (knowledge) in mathematics is 0.7, degree of truth
(knowledge) in physics is 0.6, degree of truth (knowledge) in anatomy is 0.4;
degree of truth (knowledge) in poetry is 0.9, degree of truth
(knowledge) in novels is 0.2; degree of truth (knowledge) in sculpture is 0.5.
But, according to Expert B(arbara), the truth-values of plithogenic
proposition P are: PB(0.9, 0.6, 0.2; 0.8, 0.7; 0.3).
PA /\P PB = (0.630, 0.504, 0.432; 0.720, 0.698; 0.150).
4.1. Plithogenic Probability
Since in
plithogenic probability each event E from a probability space U is characterized
by many chances of the event to occur [not only
one chance of the event E to
occur: as in classical probability, imprecise probability, and neutrosophic
probability], a plithogenic
probability distribution function, PP(x), of a
random variable x, is described by many plithogenic probability distribution
sub-functions,
where each sub-function represents the chance (with respect to a
given attribute value) that value x occurs, and these chances of
occurrence can
be represented by classical, imprecise, or neutrosophic probabilities (depending
on the type of degree of a chance).
4.2. Example of Plithogenic Probabilistic
What is the
plithogenic probability that Jennifer will graduate at the end of this semester
in her program of electrical engineering, given
that she is enrolled in and has
to pass two courses of Mathematics (Non-Linear Differential Equations, and
Stochastic Analysis), and two
courses of Mechanics (Fluid Mechanics, and Solid
Mechanics) ?
We have a 4 attribute-values of plithogenic probability.
According to her adviser, Jennifer's plithogenic single-valued fuzzy probability
of graduating at the end of this semester is:
J( 0.5, 0.6; 0.8, 0.4 ),
which means 50% chance of passing the Non-Linear Differential Equations class,
60% chance of passing the Stochastic Analysis class (as part of Mathematics),
and 80% of passing the Fluid Mechanics class,
and 40% of passing the Solid Mechanics class (as part of Physics).
Therefore the plithogenic probability in this example is composed from 4
classical probabilities.
While the Plithogenic Probability of an event E to occur is calculated with
respect to MANY chances of the event E to occur (it is calculated
with respect
to each event's attribute/parameter chance of occurrence).
Therefore, the Plithogenic Probability is a Multi-Probability (i.e.
multi-dimensional probability) -
unlike the classical, imprecise and neutrosophic probabilities that are uni-dimensional
probabilities.
The Neutrosophic Probability (and similarly for Classical Probability, and for
the Imprecise Probability) of an event E to occur is calculated
with respect to
the chance of the event E to occur (i.e. it is calculated with respect to only
ONE chance of occurrence).
5.1. Plithogenic Statistics
As
a generalization of classical statistics and neutrosophic statistics, the
Plithogenic Statistics is the analysis of events described by the
plithogenic
probability.
Plithogenic Statistics
is a MultiVariate Statistics, characterized by multiple random variables, whose
degrees may be: classical, fuzzy, intuitioninstic fuzzy, neutrosophic, or any
other fuzzy extension.
In neutrosophic
statistics we have some degree of indeterminacy, incompleteness, inconsistency
into the data or into the statistical
inference
methods.
5.2. Example of Plithogenic Statistics
Let's consider
the previous Example of Plithogenic Probability that Jenifer will graduate at
the end of this semester in her program of
electrical engineering.
Instead of defining only one probability distribution function (and drawing its
curve),
we do now 4 draw (four) probability distribution functions (and draw 4 curves),
when we consider the neutrosophic distribution as
a uni-dimensional neutrosophic function.
Therefore, Plithogenic Statistics is a multi-variate statistics.
6. Symbolic Plithogenic Algebraic Structures
Definitions of Symbolic Plithogenic Set & Symbolic
Plithogenic Algebraic Structures
Let SPS be a non-empty set, included in a
universe of discourse U, defined as follows:
SPS = {x| x = a0 + a1p1
+ a2P2 + ... + anPn, n
≥ 1, all ai in R, or C, or belong to some
given algebraic structure space, for 0 ≤ i ≤ n},
where R = the set of real numbers, C =
the set of complex numbers, and all Pi are letters (or variables), and are called Symbolic
(Literal) Plithogenic Components (Variables)}, where 1, P1, P2,
…, Pn act like a base for the elements of the above set SPS.
a0, a1, a2, …, an
are called coefficients.
SPS is called a Symbolic Plithogenic Set. And the
algebraic structures defined on this set are called Symbolic
Plithogenic Algebraic Structures.
In general, Symbolic (or Literal) Plithogenic Theory is
referring to the use of abstract symbols {i.e. the letters/parameters) P1,
P2, …, Pn, representing the plithogenic components
(variables) as above} in some theory.
References
Books
B1. Florentin Smarandache:
Plithogeny, Plithogenic
Set, Logic, Probability, and Statistics. Brussels, Belgium: Pons, 2017, 141 p.
B2. Florentin Smarandache, Mohamed Abdel-Basset (editors):
Optimization Theory
Based on Neutrosophic and Plithogenic Sets, ELSEVIER,
Academic Press, 2020, 446 p.
B3. Wilmer
Ortega Chavez, Fermin Pozo Ortega, Janett Karina Vasquez Perez, Edgar Juan Diaz
Zuniga, Alberto Rivelino Patino Rivera:
Modelo ecologico de
Bronferbrenner aplicado a la pedagogia, modelacion matematica para la toma de
decisiones bajo incertidumbre: de la logica difusa a la logica plitogenica.
NSIA Publishing House Editions, Huanuco, Peru, 2021, 144 p.
B4.
Florentin Smarandache (Special Issue Editor):
New types of
Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/Off-Set,
Neutrosophic Refined Set, and their Extension to Plithogenic
Set/Logic/Probability, with Applications. Special Issue of Symmetry (Basel, Switzerland, in Scopus, IF: 1.256),
November 2019, 714 p.,
https://www.mdpi.com/journal/symmetry/special_issues/Neutrosophic_Set_Logic_Probability.
Articles
1.
Florentin Smarandache:
Plithogenic Set, an
Extension of Crisp, Fuzzy, Intuitionistic Fuzzy, and Neutrosophic Sets -
Revisited.
Neutrosophic Sets and Systems, Vol. 21, 2018, 153-166.
2.
Florentin Smarandache:
Extension of Soft Set to
Hypersoft Set, and then to Plithogenic Hypersoft Set.
Neutrosophic Sets and Systems, Vol. 22, 2018, 168-170.
3.
Florentin Smarandache:
Conjunto plitogenico,
una extension
de los conjuntos crisp, difusos, conjuntos difusos intuicionistas y neutrosoficos
revisitado.
Neutrosophic Computing and Machine Learning, Vol. 3, 2018, 1-19.
4. Shazia
Rana, Madiha Qayyum, Muhammad Saeed, Florentin Smarandache, Bakhtawar Ali Khan:
Plithogenic Fuzzy Whole
Hypersoft Set, Construction of Operators and their Application in Frequency
Matrix Multi Attribute Decision Making Technique.
Neutrosophic Sets and Systems, Vol. 28, 2019, 34-50.
5. Nivetha
Martin, Florentin Smarandache:
Plithogenic Cognitive
Maps in Decision Making.
International Journal of Neutrosophic Science (IJNS) Vol. 9, No. 1, 2020,
9-21.
6.
Florentin Smarandache, Nivetha Martin:
Plithogenic n-Super
Hypergraph in Novel Multi-Attribute Decision Making.
International Journal of Neutrosophic Science (IJNS) Vol. 7, No. 1, 2020,
8-30.
7. Shazia
Rana, Muhammad Saeed, Midha Qayyum, Florentin Smarandache:
Plithogenic Subjective
Hyper-Super-Soft Matrices with New Definitions & Local, Global, Universal
Subjective Ranking Model.
International Journal of Neutrosophic Science (IJNS) Vol. 6, No. 2, 2020,
56-79; DOI:
10.5281/zenodo.3841624.
8. Firoz
Ahmad, Ahmad Yusuf Adhami, Florentin Smarandache:
Modified neutrosophic
fuzzy optimization model for optimal closed-loop supply chain management under
uncertainty.
Optimization Theory Based on Neutrosophic and Plithogenic Sets, 2020,
343-403; DOI:
10.1016/B978-0-12-819670-0.00015-9.
9. Sudipta
Gayen, Florentin Smarandache, Sripati Jha, Manoranjan Kumar Singh, Said Broumi,
Ranjan Kumar:
Introduction to
Plithogenic Hypersoft Subgroup.
Neutrosophic Sets and Systems, Vol. 33, 2020, 208-233.
10. Nivetha
Martin, Florentin Smarandache:
Introduction to Combined
Plithogenic Hypersoft Sets.
Neutrosophic Sets and Systems, Vol. 35, 2020, 503-510.
11. Shio
Gai Quek, Ganeshsree Selvachandran, Florentin Smarandache, J. Vimala, Son Hoang
Le, Quang-Thinh Bui, Vassilis C. Gerogiannis:
Entropy Measures for
Plithogenic Sets and Applications in Multi-Attribute Decision Making.
Mathematics 2020, 8, 965, 17 p.; DOI:
10.3390/math8060965.
12. Nivetha
Martin, Florentin Smarandache:
Concentric Plithogenic
Hypergraph based on Plithogenic Hypersoft sets - A Novel Outlook.
Neutrosophic Sets and Systems, Vol. 33, 2020, 78-91.
13. George
Bala:
Information Fusion Using
Plithogenic Set and Logic.
Acta Scientific Computer Sciences 2.7, 2020, 26-27.
14. Shawkat
Alkhazaleh:
Plithogenic Soft Set.
Neutrosophic Sets and Systems, Vol. 33, 2020, 256-274.
15. R.
Sujatha, S. Poomagal, G. Kuppuswami, Said Broumi:
An Analysis on Novel
Corona Virus by a Plithogenic Fuzzy Cognitive Map Approach.
International Journal of Neutrosophic Science (IJNS), Volume 11, Issue 2,
2020, 62-75; DOI: 10.5281/zenodo.4275788.
16. S. P.
Priyadharshini, F. Nirmala Irudayam, F. Smarandache:
Plithogenic Cubic Sets.
International Journal of Neutrosophic Science (IJNS), Volume 11, Issue 1,
2020, 30-38; DOI: 10.5281/zenodo.4275725.
17. Prem
Kumar Singh:
Plithogenic set for
multi-variable data analysis.
International Journal of Neutrosophic Science (IJNS), Volume 1, Issue 2,
2020, 81-89; DOI: 10.5281/zenodo.3988028.
18. C.
Sankar, R. Sujatha, D. Nagarajan:
TOPSIS by Using
Plithogenic Set in COVID-19 Decision Making.
International Journal of Neutrosophic Science (IJNS), Volume 10, Issue 2,
2020, 116-125; DOI: 10.5281/zenodo.4277255.
19. Nivetha
Martin, R. Priya:
New Plithogenic sub cognitive maps approach with mediating effects of factors in
COVID-19 diagnostic model.
Journal of Fuzzy
Extension & Applications
(JFEA), Volume 2, Issue 1, Winter 2021, 1-15; DOI:
10.22105/JFEA.2020.250164.1015.
20. Mohamed Abdel-Basset, Rehab Mohamed, Florentin Smarandache, Mohamed Elhoseny: A
New Decision-Making Model Based on
Plithogenic Set for Supplier Selection. Computers, Materials
& Continua, 2021, vol. 66, no. 3, 2752-2769. DOI:10.32604/cmc.2021.013092
21. Nivetha Martin, Florentin Smarandache, R. Priya: Introduction
to Plithogenic Sociogram with preference representations by Plithogenic
Number. Journal
of Fuzzy Extension & Applications, 15 p.
22. S.P. Priyadharshini, F. Nirmala Irudayam, F. Smarandache: Plithogenic
Cubic Set. International Journal of Neutrosophic Science (IJNS),
2020, Vol. 11, No. 1, 30-38.
23.
Muhammad Rayees Ahmad, Muhammad Saeed , Usman Afzal, Miin-Shen Yang: A
Novel MCDM Method Based on Plithogenic Hypersoft
Sets under Fuzzy Neutrosophic
Environment. Symmetry 2020, 12, 1855, 23 p.; DOI:
10.3390/sym12111855
24.
Korucuk Selcuk, Demir Ezgi, Karamasa Caglar, Stevic Zeljko: Determining
The Dimensions of The Innovation Ability in Logistics Sector
by Using Plithogenic-Critic Method: An Application in Sakarya Province. International
Review, No. 1-2, September 2020, 10 p.
25.
Rawa Alwadani, Nelson Oly Ndubisi: Family
business goal, sustainable supply chain management, and platform economy: a
theory
based review & propositions for future research. International
Journal of Logistics Research and Applications, 25 p.; DOI:
10.1080/13675567.2021.1944069
26.
S.P. Priyadharshini, F. Nirmala Irudayam: A
Novel Approach of Refined Plithogenic Neutrosophic Sets in Multi Criteria
Decision
Making. International Research Journal of Modernization in
Engineering Technology and Science, Volume 3, Issue 4, May 2021, 5 p.
27.
Alptekin
Uluta¸ Ayse Topal, Darjan Karabasevic, Dragisa Stanujkic, Gabrijela Popovic, and
Florentin Smarandache, Prioritization of
Logistics
Risks with Plithogenic PIPRECIA Method, in C. Kahraman et al. (Eds.): INFUS
2021, Springer, LNNS 308, pp. 663–670, 2022.
28. Florentin Smarandache,
Introduction to the
Symbolic Plithogenic Algebraic Structures (revisited), Neutrosophic
Sets and Systems, Vol. 53, 2022.
29. Florentin Smarandache, An Overview of
Plithogenic Set and Symbolic Plithogenic Algebraic Structures
(Review Paper), J. Fuzzy. Ext.
Appl. Vol. 4, No. 1 (2023) 48–55,
http://fs.unm.edu/P/Plithogeny-JFEA.pdf
30. W.B. Vasantha Kandasamy, K.
Ilanthenral, F. Smarandache, Plithogenic Graphs, EuropaNova, Belgium, 2020.