**NEW TYPES OF SOFT SETS:
**

**HyperSoft Set, **

**IndetermSoft Set, IndetermHyperSoft Set,
SuperHyperSoft Set, TreeSoft Set **

**(An Improved Version)**

The Soft Set was introduced by Molodtsov
[1] in
1999.

Further on, *four new types of Soft
Sets*, such as: the HyperSoft Set (2018), IndetermSoft Set (2022), IndetermHyperSoft Set (2022), and TreeSoft Set (2022),
were introduced by Smarandache [2-7].

Let's recall their definitions
together with real examples.

** Definition of Soft Set**

Let U be a universe of discourse, P(U)
the power set of U, and A a set of attributes. Then, the pair (F, U), where F: A
→ P(U) is called a Soft Set over U.

**Real Example of
Soft Set**

Let U = {Helen, George, Mary,
Richard} and a set M = {Helen, Mary, Richard} included in U.

Let the attribute be: a = size, and
its attribute’ values respectively:

Size = A_{1} = {small,
medium, tall},

Let the function be: F: A_{1}
→ P(U).

Then, for example:

F(tall) = {Helen, Mary}, which means
that both, Helen and Mary, are tall.

** 1. Definition of IndetermSoft Set **

Let *U* be a universe of discourse,
*H* a
non-empty subset of *U*, and *P(H)* be the powerset of *H*. Let *a* be an
attribute, and *A* be a set of this attribute-values. Then *F: A → P(H)*
is called an IndetermSoft Set if at least one of the bellow occurs:

i) the set *A* has some indeterminacy;

ii) at least one of the sets *H* or *P(H)* has some indeterminacy;

iii) the function* F* has some
indeterminacy, i.e. there exist at least one relationship *F(a) = M*, such
that at least one of* a* or *M* has some indeterminacy (not unique,
unclear, incomplete, unknown, etc.).

IndetermSoft Set, as extension of the
classical (determinate) Soft Set, deals with indeterminate data, because there
are sources unable to provide exact or complete information on the sets A, H, or
P(H), nor on the function F. We did not add any indeterminacy, we found the
indeterminacy in our real world. Because many sources give
approximate/uncertain/incomplete/conflicting information, not exact information
as in the Soft Set, as such we still need to deal with such situations.

Herein, ‘Indeterm’ stands for
‘Indeterminate’ (uncertain, conflicting, incomplete, not unique outcome, unknown).

Similarly, distinctions between
determinate and indeterminate operators are taken into consideration.
Afterwards, an IndetermSoft Algebra is built, using a determinate soft operator
(joinAND), and three indeterminate soft operators (disjoinOR, exclussiveOR,
NOT), whose properties are further on studied.

Smarandache has generalized the Soft
Set to the HyperSoft set by transforming the function F into a multi-attribute
function, and then he introduce the hybrids of Crisp, Fuzzy,
Intuitionistic
Fuzzy, Neutrosophic,
other fuzzy extensions, and Plithogenic
HyperSoft Set.

The classical Soft Set is based on a
determinate function (whose values are certain, and unique), but in our world
there are many sources that, because of lack of information or ignorance,
provide indeterminate (uncertain, and not unique – but hesitant or alternative)
information. They can be modeled by operators having some degree of
indeterminacy due to the imprecision of our world.

** Real Example of IndetermSoft Set**

Assume a town has many houses.

1) Indeterminacy with respect to the
set A of attributes.

You ask the source:

- What are all colors of the houses?

The source: I know for sure that
there are houses of colors red, yellow, and blue, but I do not know if there are
houses of other colors (?)

2) Indeterminacy with respect to the
set H of houses.

You ask the source:

- How many houses are in the town?

The source:

- I never counted them, but I
estimate their number to be between 100-120 houses.

3) Indeterminacy with respect to the
function.

3a) You ask a source:

- What houses have the red color in
the town?

The source:

- I am not sure, I think the houses h_{1}
or h_{2}.

Therefore, F(red) = h_{1} or
h_{2} (indeterminate / uncertain answer).

3b) You ask again:

- But, what houses are yellow?

The source: - I do not know, the only
thing I know is that the house h_{5} is not yellow because I have
visited it.

Therefore, F(yellow) = not h_{5}
(again indeterminate / uncertain answer).

3c) Another question you ask:

- Then what houses are blue?

The source:

- For sure, either h_{8} or h_{9}.

Therefore, F(blue) = either h_{8}
or h_{9} (again indeterminate / uncertain answer).

** **This is the IndetermSoft Set.

** 2. Definition of HyperSoft Set**

The soft set was extended to the
hypersoft set by transforming the function F into a multi-attribute function.
Afterwards, the hybrids of HyperSoft Set with the Crisp, Fuzzy, Intuitionistic
Fuzzy, Neutrosophic, other fuzzy extensions, and Plithogenic Set were
introduced.

Let U be a universe of discourse, P(U)
the power set of U. Let a_{1} , a_{2} , … , a_{n}, for n
≥ 1, be n distinct attributes, whose corresponding attribute values are
respectively the sets A_{1} , A_{2} , … , A_{n}, with A_{i}
∩ A_{j} = Φ, for i ≠ j, and i,j in {1,
2, … , n}. Then the pair (F, A_{1} × A_{2 }× … × A_{n}
), where
F: A_{1} × A_{2} × … × A_{n} → P(U), is called a
HyperSoft Set over U.

**Real Example of
HyperSoft Set**

Let U = {Helen, George, Mary,
Richard} and a set M = {Helen, Mary, Richard} included in U.

Let the attributes be: a_{1}
= size, a_{2} = color, a_{3} = gender, a_{4} =
nationality, and their attributes’ values respectively:

Size = A_{1} = {small,
medium, tall},

Color = A_{2} = {white,
yellow, red, black},

Gender = A_{3} = {male,
female},

Nationality = A_{4} =
{American, French, Spanish, Italian, Chinese}.

Let the function be: F: A_{1}
× A_{2} × A_{3} × A_{4} → P(U).

Then, for example:

F({tall, white, female, Italian}) =
{Helen, Mary}, which means that both, Helen and Mary, are tall and white and
female and Italian.

Notice that this is an extension of
the previous Real Example of Soft Set.

** 3. Definition of IndetermHyperSoft
Set**

Let U be a universe of discourse, H a
non-empty subset of U, and P(H) the powerset of H. Let a_{1} , a_{2}
, … , a_{n}, for n ≥ 1, be n distinct attributes, whose corresponding
attribute-values are respectively the sets A_{1} , A_{2} , … , A_{n},
with A_{i} ∩ A_{j} = Φ for i ≠
j, and i, j in {1, 2, … , n}. Then the pair
(F, A_{1} × A_{2 }× … × A_{n} ), where F: A_{1}
× A_{2} × … × A_{n} → P(H), is called an IndetermHyperSoft
Set over U if at least one of the bellow occurs:

i) at least one of the sets A_{1}
, A_{2} , … , A_{n} has some indeterminacy;

ii) at least one of the sets H or P(H) has some
indeterminacy;

iii) the function F has some
indeterminacy, i.e. there exist at least a relationship F(e_{1}, e_{2, }…,
e_{n}) = M such that at least one of e_{1}, a_{2, }…, e_{n},
or M is indeterminate (unclear, uncertain, conflicting, or not unique, unknown,
etc.).

Similarly, IndetermHyperSoft Set is an extension of the HyperSoft
Set, when there is indeterminate data, or indeterminate function, or
indeterminate sets.

** Real Example of IndetermHyperSoft
Set**

Assume a town has many houses.

1) Indeterminacy with respect to the
product set A_{1} × A_{2 }× … × A_{n} of attributes.

You ask the source:

- What are all colors and sizes of
the houses?

The source: I know for sure that
there are houses of colors of red, yellow, and blue, but I do not know if there
are houses of other colors (?) About the size, I saw many houses that are small,
but I do not remember to have seeing big houses.

2) Indeterminacy with respect to the
set H of houses.

You ask the source:

- How many houses are in the town?

The source:

- I never counted them,
but I estimate their number to be between 100-120 houses.

3) Indeterminacy with respect to the
function.

3a) You ask a source:

- What houses are of red color and
big size in the town?

The source:

- I am not sure, I think the houses
h_{1} or h_{2}.

Therefore, F(red, big) = h_{1}
or h_{2} (indeterminate / uncertain answer).

3b) You ask again:

- But, what houses are yellow and
small?

The source:

- I do not know, the only thing I
know is that the house h_{5} is neither yellow nor small because I have
visited it.

Therefore, F(yellow, small) = not h_{5}
(again indeterminate / uncertain answer).

3c) Another question you ask:

- Then what houses are blue and big?

The source:

- For sure, either h_{8} or h_{9.}

Therefore, F(blue, big) = either h_{8}
or h_{9} (again indeterminate / uncertain answer).

This is the IndetermHyperSoft Set.

**4. Definition
of SuperHyperSoft Set**

The SuperHyperSoft Set is an
extension of the HyperSoft Set.

As for the SuperHyperAlgebra,
SuperHyperGraph, SuperHyperTopology and in general for SuperHyperStructure and
Neutrosophic SuperHyperStructure (that includes indeterminacy) in any field of
knowledge, “Super” stands for working on the powersets (instead of sets) of the
attribute value sets.

Let U be a universe of discourse, P(U)
the power set of U. Let a_{1} , a_{2} , … , a_{n}, for n
≥ 1, be n distinct attributes, whose corresponding attribute values are
respectively the sets A_{1} , A_{2} , … , A_{n}, with A_{i}
∩ A_{j} = Φ, for i ≠ j, and i,j in {1,
2, … , n}. Let P(A_{1}) , P(A_{2}) , … , P(A_{n}) be the
powersets of of the sets A_{1} , A_{2} , … , A_{n}
respectively.

Then the pair ( F, P(A_{1}) ×
P(A_{2}) × … × P(A_{n})
), where F: P(A_{1}) × P(A_{2}) × … × P(A_{n}) → P(U),
where × means Cartesian product, is called a SuperHyperSoft Set over U.

**Real Example of
SuperHyperSoft Set**

If we define the function: F: P(A_{1})
× P(A_{2}) × P(A_{3}) × P(A_{4}) → P(U), we
get a SuperHyperSoft Set.

Let’s assume, from the previous
example, that:

F({medium,tall},{white, red,
black},{female},{American, Italian}) = {x_{1} , x_{2}} , which
means that:

F( {medium or tall} and {white or red
or black} and {female} and {American or Italian} ) = {x_{1} , x_{2}}.

Therefore, the SuperHyperSoft Set
offers a larger variety of selections, so x_{1} and x_{1} may
be:

either medium, or tall (but not
small),

either white, or red, or black (but
not yellow),

mandatory female (not male),

and either American, or Italian (but
not French, Spanish, Chinese).

**5. Definition of TreeSoft Set**

Let U be a universe of discourse, and
H a non-empty subset of U, with P(H) the powerset of H.

Let A be a set of attributes
(parameters, factors, etc.),

A= {A_{1} , A_{2} , … , A_{n}},
for integer n ≥ 1, where A_{1} , A_{2} , … , A_{n }
are considered_{ }__attributes of first level__ (since they have
one-digit indexes).

Each attribute A_{i}, 1 ≤ i ≤
n, is formed by sub-attributes:

A_{1}= {A_{1,1} , A_{1,2} , … }

A_{2}= {A_{2,1} , A_{2,2} , … }

.........................

A_{n}= {A_{n,1} , A_{n,2} , … }

where the above A_{i,j are sub-attributes (or attributes
of second level) (since they have two-digit indexes).}

Again, each sub-attribute A_{i,j}
is formed by sub-sub-attributes (or __attributes of third level__):

A_{i,j,k}

And so on, as much refinement as
needed into each application, up to sub-sub-…-sub-attributes (or __attributes
of m-level__ (or having *m* digits into the indexes):

A_{i1,i2,...,im}

Therefore, a graph-tree is formed,
that we denote as Tree(A), whose root is A (considered of __level zero__),
then nodes of __level 1__, __level 2__, up to __level m__.

We call *leaves* of the
graph-tree, all terminal nodes (nodes that have no descendants).

Then the TreeSoft Set is:

F: P(Tree(A)) → P(H)

Tree(A) is the set of all nodes and
leaves (from level 1 to level m) of the graph-tree, and P(Tree(A)) is the
powerset of the Tree(A).

All node sets of the *TreeSoft Set
of level m* are:

Tree(A) = {A_{i1}| i_{1}=
1, 2, ... }

The first set is formed by the nodes
of level 1, second set by the nodes of level 2, third set by the nodes of level
3, and so on, the last set is formed by the nodes of level m. If the graph-tree
has only two levels (m = 2), then the TreeSoft Set is reduced to a MultiSoft Set
[8].

**Practical
Example of TreeSoft Set of Level 3 **

This is a classical tree,
whose:

Level 0 (the root) is the node
Attributes;

Level 1 is formed by the
nodes: Size, Location;

Level 2 is formed by the nodes
Small, Big, Arizona, California;

Level 3 is formed by the nodes
Phoenix, Tucson.

Let’s consider *H =
{h*_{1}, h_{2}, ..., h_{10}} be a set of houses, and
*P(H)* the powerset of *H*.

And the set of Attributes: *
A = {A*_{1}, A_{2}}, where *A*_{1} = Size, A_{2}
= Location.

Then *A*_{1} = {A_{11},
A_{12}} = {Small, Big}, *A*_{2} = {A_{21}, A_{22}}
= {Arizona, California} as American states.

Further
on, *A*_{22} = {A_{211}, A_{212}} = {Phoenix, Tucson}
as Arizonian cities

Let’s
assume that the function *F* gets the following values:

*F(Big, Arizona, Phoenix} = { h*_{9}, h_{10
}}

*F(Big, Arizona, Tucson) = { h*_{1}, h_{2},
h_{3}, h_{4 }}

*F(Big,
Arizona)* = all big
houses from both cities, Phoenix and Tucson

=
*F(Big, Arizona, Phoenix) **
∪ F(Big, Arizona,
Tucson) = { h*_{1}, h_{2}, h_{3}, h_{4,
}h_{9, }h_{10 }}.

**Conclusion**

The HyperSoft Set (2018) is a generalization of Soft Set (1999),
from a uni-variate function to a multi-variate function F;

IndetermSoft Set (2022) is an extension of the Soft Set, from
the determinate data to indeterminate data;

IndetermHyperSoft Set (2022) is an extension of the HyperSoft
Set, from the determinate data to indeterminate data;

SuperHyperSoft Set (2023) is a generalization of the HyperSoft
Set, where one considers the powersets of the attribute value sets;

and TreeSoft Set (2022)
is a generalization of the MultiSoft Set that is a subclass of the HyperSoft
Set.

**References**

1. Molodtsov, D. (1999) Soft Set
Theory First Results. Computer Math. Applic. 37, 19-31

2. F. Smarandache, Extension of Soft
Set to Hypersoft Set, and then to Plithogenic Hypersoft Set, Neutrosophic Sets
and Systems, vol. 22, 2018, pp. 168-170 DOI: 10.5281/zenodo.2159754;
http://fs.unm.edu/NSS/ExtensionOfSoftSetToHypersoftSet.pdf

3. Florentin Smarandache, Extension
of Soft Set to Hypersoft Set, and then to Plithogenic Hypersoft Set (revisited),
Octogon Mathematical Magazine, vol. 27, no. 1, April 2019, pp. 413-418

4. Florentin Smarandache,
Introduction to the IndetermSoft Set and IndetermHyperSoft Set, Neutrosophic
Sets and Systems, Vol. 50, pp. 629-650, 2022 DOI: 10.5281/zenodo.6774960;
http://fs.unm.edu/NSS/IndetermSoftIndetermHyperSoft38.pdf

5. F. Smarandache, (2015).
Neutrosophic Function, in Neutrosophic Precalculus and Neutrosophic
Calculus, EuropaNova, Brussels, 14-15, 2015;
http://fs.unm.edu/NeutrosophicPrecalculusCalculus.pdf

6. F. Smarandache, Neutrosophic
Function, in Introduction to Neutrosophic Statistics, Sitech & Education
Publishing, 74-75, 2014;
http://fs.unm.edu/NeutrosophicStatistics.pdf

7. F. Smarandache, Soft Set Product
extended to HyperSoft Set and IndetermSoft Set Product extended to
IndetermHyperSoft Set, Journal of Fuzzy Extension and Applications, 2022, DOI:
10.22105/jfea.2022.363269.1232,
http://www.journal-fea.com/article_157982.htm

8. Shawkat Alkhazaleh, Abdul Razak Salleh, Nasruddin Hassan, Abd Ghafur
Ahmad, Multisoft Sets, Proc. 2^{nd} International Conference on
Mathematical Sciences, pp. 910-917, Kuala Lumpur, Malaysia, 2010.

9. F. Smarandache,
New Types of Soft Sets: HyperSoft Set, IndetermSoft Set, IndetermHyperSoft Set,
and TreeSoft Set*, *International
Journal of Neutrosophic Science, Vol.
20 , No.
4, (2023) :
58-64,
http://fs.unm.edu/TSS/NewTypesSoftSets-IJNS.pdf

10. Florentin
Smarandache, New Types of Soft Sets ”HyperSoft Set, IndetermSoft
Set, IndetermHyperSoft Set, and TreeSoft Set”: An Improved
Version, Neutrosophic Systems with Applications, 35-41, Vol. 8, 2023,
http://fs.unm.edu/TSS/NewTypesSoftSets-Improved.pdf.

11.
Florentin Smarandache,
Foundation of the SuperHyperSoft Set and the Fuzzy Extension
SuperHyperSoft Set: A New Vision, Neutrosophic Systems with Applications, Vol.
11, 48-51, 2023, Neutrosophic Systems with Applications, Vol. 11, 48-51, 2023,
http://fs.unm.edu/TSS/SuperHyperSoftSet.pdf.