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: . Brussels, Belgium: Pons, 2017, 141 p.

B2. Florentin Smarandache, Mohamed Abdel-Basset (editors): , 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: . NSIA Publishing House Editions, Huanuco, Peru, 2021, 144 p.

B4. Florentin Smarandache (Special Issue Editor): . Special Issue of Symmetry (Basel, Switzerland, in Scopus, IF: 1.256), November 2019, 714 p., .

Articles

1. Florentin Smarandache: . Neutrosophic Sets and Systems, Vol. 21, 2018, 153-166.

2. Florentin Smarandache: . Neutrosophic Sets and Systems, Vol. 22, 2018, 168-170.

3. Florentin Smarandache: . Neutrosophic Computing and Machine Learning, Vol. 3, 2018, 1-19.

4. Shazia Rana, Madiha Qayyum, Muhammad Saeed, Florentin Smarandache, Bakhtawar Ali Khan: . Neutrosophic Sets and Systems, Vol. 28, 2019, 34-50.

5. Nivetha Martin, Florentin Smarandache: . International Journal of Neutrosophic Science (IJNS) Vol. 9, No. 1, 2020, 9-21.

6. Florentin Smarandache, Nivetha Martin: . International Journal of Neutrosophic Science (IJNS) Vol. 7, No. 1, 2020, 8-30.

7. Shazia Rana, Muhammad Saeed, Midha Qayyum, Florentin Smarandache: . International Journal of Neutrosophic Science (IJNS) Vol. 6, No. 2, 2020, 56-79; DOI: .

8. Firoz Ahmad, Ahmad Yusuf Adhami, Florentin Smarandache: . Optimization Theory Based on Neutrosophic and Plithogenic Sets, 2020, 343-403; DOI: .

9. Sudipta Gayen, Florentin Smarandache, Sripati Jha, Manoranjan Kumar Singh, Said Broumi, Ranjan Kumar: . Neutrosophic Sets and Systems, Vol. 33, 2020, 208-233.

10. Nivetha Martin, Florentin Smarandache: . 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: . Mathematics 2020, 8, 965, 17 p.; DOI: .

12. Nivetha Martin, Florentin Smarandache: . Neutrosophic Sets and Systems, Vol. 33, 2020, 78-91.

13. George Bala: . Acta Scientific Computer Sciences 2.7, 2020, 26-27.

14. Shawkat Alkhazaleh: . Neutrosophic Sets and Systems, Vol. 33, 2020, 256-274.

15. R. Sujatha, S. Poomagal, G. Kuppuswami, Said Broumi: . 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: . International Journal of Neutrosophic Science (IJNS), Volume 11, Issue 1, 2020, 30-38; DOI: 10.5281/zenodo.4275725.

17. Prem Kumar Singh: . 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: . 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 SelectionComputers, 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

NumberJournal 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 EnvironmentSymmetry 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 ProvinceInternational 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 researchInternational 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

MakingInternational 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

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