Introduction to the Symbolic Plithogenic Algebraic Structures (revisited)

In this paper, we recall and study the new type of algebraic structures called Symbolic Plithogenic Algebraic Structures. Their operations are given under the Absorbance Law and the Prevalence Order.


Hybridization of Classical, Fuzzy, and Fuzzy Extension Sets
The real applications require many times to deal with multiple types of classical, fuzzy, and fuzzy extension sets.
Assume that, starting from a neutrosopohic element of the form x (T, I The hybridization chain may be as long as needed, and may deal with various types of classical, fuzzy, and fuzzy extension sets -including repeated types.

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: 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.

Definition of Plithogenic Numbers (PN)
The numbers of the form

Prevalence Order (PO)
The experts establish a prevalence order [1], or total order, according to the importance of each attribute/parameter (Pi) into the application. To obtain a total order among the symbolic plithogenic components 1 2 { , ,..., } n P P P , one defines some relationships (laws) between them. The most used one is the absorbance law.

Absorbance Law
We recall and use now our 2015 Absorbance Law [1], simply defined as: the greater absorbs the smaller [the bigger fish eats the smaller fish].

Multiplication and Power of Symbolic Plithogenic Components under the Absorbance Law
We assume that in the above definition of the plithogenic numbers, the symbolic plithogenic components are ranked increasingly, or 12 ... n P P P    (prevalence order) where "<" may signify: smaller, less important, under, inferior, etc. Whence, the multiplication and power of symbolic plithogenic components are: In general,

Example of Type-2 Plithogenic Set
Assume that Expert 2 is not totally confident on the evaluation of the Expert 1 in the above example. Thus, he decides to evaluate the first evaluation. Expert 2 may, as well, use any types of degrees -according to the expert desire and tools, not necessarily the same as in the previous evaluation. For the sake of simplicity, let's consider that Expert 2 also uses fuzzy degrees. Now one gets a Type-2 Plithogenic Set: And similarly for George's and Mary's second round of degrees.

Example of Type-3 Plithogenic Set
The process may go on and have an Expert 3 evaluate the Expert 2. Assume Expert 3 uses neutrosophic degrees. One may generalize to Type-k Plithogenic Set, recurrently going on from a type to the next type, but it becomes more sophisticated and not usable in practice.

Example of Symbolic Plithogenic Numbers
The corresponding Symbolic Plithogenic Algebraic Structure is based on the symbolic (or literal) plithogenic components W, T, O, B, H, and we get the plithogenic numbers (PN) of the form: PN = a + bW + cT + dO + eB + fH, where a, b, c, d, e, f are real, or complex numbers, or they may belong to a set of a given classical algebraic structure. As a particular example, let PN1 = 2 -3W + 5T -O + 6B -4H.
In this example, let's assume that the prevalence order is: W < T < O < B < H, where "<" means "less important", or W is less important than T, which is less important than O, which is less important that B, which is less important than H.
The absorbance law is defined as follows: the most important absorbs the less important in the multiplication operation, for example W T T =, since T absorbs W because T is more important (bigger) than W. Similarly for the other multiplications.

Operations with Plithogenic Numbers
Let's consider two plithogenic numbers: , where  is the classical multiplication, as in classical algebra, using the above multiplication of symbolic plithogenic components.

Scalar Multiplication of Plithogenic Numbers
As particular case: 00 i P = .
( , , ) SPS + is a Symbolic Plithogenic Commutative Ring, with the plithogenic unitary element: The symbolic plithogenic components Pi's are not inversible, therefore the elements of SPS are non-inversible (except the plithogenic unitary element * 1 ).
The negative power of a plithogenic number 1 () m PN − does not exist.

Alternative Multiplication of Plithogenic Numbers
SPS + , is a Symbolic Plithogenic Commutative Ring, with the unitary element: The plithogenic numbers that have coefficients equal to zero do not have an inverse, for example: 2 + 3P1 -5P3 = 2 + 3P1 + 0P2 -5P3 is not inversible. There are j-tuple infinities of quotients when i > j, also i-tuple infinities of quotients when i = j, and no quotient (indeterminate division) when i < j. Therefore, the operation of division d(. , .) of symbolic plithogenic components  x P x x P x P x P x P x P P

Division of Symbolic Plithogenic Components
x P x PP x P P x P P x P P x P P PP  , which is impossible.
This multiplication, P4 times any of 1, P1, P2, …, Pn, will give a result that is greater than or equal to P4 according to the absorbing law.
This division is undefined (indeterminate).

Division of Symbolic Plithogenic Numbers
Let consider two symbolic plithogenic numbers as below: where r, s ≤ n are positive integers, and the leading coefficients (the coefficients of the highest/largest symbolic plithogenic components Pr and respectively Ps) are nonnull, 0, 0 The division is also based on the absorbance law.
There is only one quotient (solution): Let's check the result:

Replace it into the first equation:
This is also a NeutroOperation since one has indeterminate cases. Indeterminacy.

Florentin Smarandache, Introduction to the Symbolic Plithogenic Algebraic Structures (revisited)
0 -x3 = 5, then x3 = -5. x are coefficients that we need to find out. After raising to the power k the right-hand side, we identify the coefficients two by two. , where we need to find x0 and x1.
Raise both sides to the second power: Identify the coefficients:

15.9.
Remark 1 Other operations may be constructed on the Symbolic Plithogenic Set (SPS), giving birth to various symbolic plithogenic algebraic structures.

Remark 2
All previous operations are valid for the absorbance law and prevalence order defined above. If different law and order are defined by the experts, then different operations and results one gets.

Neutrosophic Quadruple Numbers
Let's consider an entity (i.e. a number, an idea, an object, etc.) which is represented by a known part (a) and an unknown part ( + + ). Numbers of the form = + + + , where a, b, c, d are real (or complex) numbers (or intervals, or in general subsets), and T = truth / membership / probability, I = indeterminacy / neutrality, F = false / membership / improbability, are called Neutrosophic Quadruple (Real respectively Complex) Numbers (or Intervals, or in general Subsets) [1].
"a" is called the known part of NQ, while " + + " is called the unknown part of NQ. Neutrosophic Quadruple Numbers [1] are particular case of the Plithogenic Numbers, since one takes n = 3, and P1, P2, P3 are more general than T, I, and F respectively.
Refined Neutrosophic Quadruple Numbers are also particular case of the Plithogenic Numbers, since instead of symbolic sub-truths / sub-indeterminacies / sub-falsehoods ,,
According to him, Turiyam component (Y) means: "Rejection of both acceptation and rejection of attribute at the given time i.e. unknown region (l). It needs Turiyam consciousness to explore it" [9].
Turiyam Set is very similar to Belnap's Logic, based on: True (T), False (F), Unknown (U), and Contradiction (C), where T, F, U, C are taken as symbols, not numbers. Belnap's Logic is a particular case of Refined Neutrosophic Logic [10].
Turiyam Set was defined as:

STN a a T a F a I a Y
= + + + + where ∈ . It is clear that Turiyam Set (2021) is a particular case of the Plithogenic Set, because one replaces n = 4, and P1, P2, P3, P4 by T, F, I, Y respectively, since the symbolic plithogenic components may be either independent, or dependent, or partially independent/dependent as we desire.
The operations on TS were defined as particular cases to Smarandache's 2015 neutrosophic quadruple numbers and absorbance law [1] and 2019 symbolic plithogenic numbers [5]. Let   0 1  2  3  4  0  1  2  3  4 ( , , , , ) x a a T a F a I a Y a a T a F a I  ( , , , , )  y a b a b T a b F a b I a b Y   a b  a b T a b F a b I

T T T T F F F F I I I I Y Y Y Y T Y Y T Y T F F T F T I I T I I Y Y I I F Y Y F Y F I I F I
 = =  = =  = =  = =  =  =  =  =  =  =  =  =  =  =  =  = While using the absorbance law (the stronger absorbs the weaker) and the prevalence order T < F < I < Y (as chosen by author Singh [12]) it would have been much simpler.

y a a T a F a I a Y b bT b F b I b Y
 = + + + +  + + + + Then similarly multiply them term by term, taking into consideration the multiplication of symbolic components T, F, I, Y as explained above.
Scalar Multiplication in the similar way: x c a a T a F a I a Y c a c a T c a F

Practical Application
Since the cases n = 3 and 4 of Symbolic Plithogenic Algebraic Structures have been investigated, the reader may try to develop it for the case when n = 5, using Hexagonal Plithogenic Numbers (HPN), hexa since the dimension of HPN is 5 + 1 = 6 because one has 6 vectors into the base: 1  2  3  4  5 1, , , , , P P P P P . HPN a a P a P a P a P a P = + + + + + , where all coefficients ai belong to a given set.
As practical application, for example, assume that the parameters represent various colors, C1, C2, C3, C4, C5, then we denote it as: HPN a a C a C a C a C a C = + + + + + .
As multiplication law of the symbolic plithogenic components Ci with Cj one adopts a law from the real world. For example, if C1 = yellow, and C2 = red, then it makes sense to consider 12 CC  = pink (because yellow mixed with red give pink), and so on. In this practical application, the absorbance law does not work, that's why one designs a new law in order to be able to multiply the components.

Open Question
Florentin Smarandache, Introduction to the Symbolic Plithogenic Algebraic Structures (revisited) Future possible study for researchers would be to investigate the infinite-case, we mean when each element in the plithogenic set (section 2 above) is characterized by infinitely many attributes (parameters), and similarly the symbolic plithogenic numbers (section 3 above) have infinitely many symbolic plithogenic components 12 , ,..., P P P  and, eventually, their applications.

Conclusion
In this paper, the new types of algebraic structures from 2018-2019, called Symbolic Plithogenic Algebraic Structures, were revisited, and afterwards compared to other related structures.
We proved that the Symbolic Plithogenic Numbers are generalizations of Neutrosophic Quadruple Numbers, Refined Neutrosophic Quadruple Numbers, and Symbolic Turiyam Numbers.
Consequently, the Symbolic Plithogenic algebraic structures (semigroup, group, ring, etc.) are generalization of the corresponding algebraic structures built on these particular cases described above. We recalled the Symbolic Plithogenic Group and Ring.
Many examples and practical applications were also revealed. Any future application may require a special multiplication law of the components and of plithogenic numbers that the experts should design themselves.