Definition 1 (see [16].)Assume the set $$ and the order relation $$ stated by $$ if and only if $$ for all $$, then $$ forms a complete lattice. An ivf-set $$ over $$ is characterized by mapping $$, where $$ is called membership function of $$. The collection of all ivf-sets over $$ is represented by Ω_ivfs.

Definition 2 (see [1].)Let $$ be an initial universe and $$ be a set of parameters then a SAAF is a mapping $$ and defined as $$, where $$ denotes the power set of $$, $$ with k ≤ n. The pair $$ is known as s-set and represented by $$. The subsets $$ are known as $$ -approximate sets having all $$ -approximate elements. The pair $$ is called soft-universe. The collection of s-sets is denoted by Ω_ss.

Definition 3 (see [17].)For any soft-universe $$ with $$, an ivfs-set $$ is characterized by mapping $$, where $$ is the same as stated in Definition 2, and $$ is known as SAAF of $$. The collection of ivfs-sets is denoted by Ω_ivfss.

Definition 4 (see [15].)Let $$, then their soft product operations, i.e., ∧&∨ are given as:

• (1)

  The ∧-product (AND-operation) of $$ and $$ is an ivfs-set defined by

• (2)

  The ∨-product (OR-operation) of $$ and $$ is an ivfs-set defined by

The following two soft-inclusions relations Jun’s inclusion $$ in [17] and Liu’s inclusion $$ in [13] are prominent in literature for understanding the set-theoretic operations of ivfs-sets.

Definition 5 (see [17].)Let $$, then

• (1)

  $$ is said to be ivfs $$-subset of $$, denoted by $$, if for every $$ such that $$

• (2)

  $$ and $$ are said to be ivfs $$-equal, denoted by $$, if $$ and $$

Definition 6 (see [13].)Let $$, then

• (1)

  $$ is said to be ivfs $$-subset of $$, denoted by $$, if for every $$ such that $$

• (2)

  $$ and $$ are said to be ivfs $$-equal, denoted by $$, if $$ and $$

Note: both $$ and $$ are termed as ivfs $$-inclusion and ivfs $$-inclusion respectively.

Proposition 7. If $$, then it implies $$.

Definition 8 (see [13].)$$ is said to be identical to $$, denoted by $$, if $$ and $$ for every $$.

Proposition 9. If $$, then it implies $$ which further implies $$.

Definition 10 (see [23].)Let $$ be an initial universe and $$ be a set of parameters. The respective attribute-valued nonoverlapping sets of each element of $$ are $$, $$, $$, ….., $$ and $$, where each $$ is a n-tuple element of $$ and $$, |•| denotes set cardinality, then a MAAF is a mapping $$ and defined as $$, where $$ denotes the power set of $$, $$ with k ≤ r. The pair $$ is known as hs-set and represented by $$. The collection of all hs-sets is symbolized as Ω_hss.

Definition 11 (see [23].)If $$ be the collection of all fuzzy sets, then a hs-set $$ is said to be fhs-set if $$, where $$ is same as discussed in Definition 10, and $$ is an approximate element of fhs-set for $$.

Definition 12. Let $$ be an initial universe and $$ be a set of parameters. The respective subparametric-valued disjoint sets are $$, $$, $$,….., $$,and $$, where each $$ is a n-tuple element of $$ and $$, |•| denotes set cardinality then the pair $$ is known as ivfhs-set, where $$ and defined as $$, and $$ with k ≤ s. The collection of all ivfhs-sets is symbolized as $$.

Example 1. Suppose an organization plans to recruit a candidate to fill a vacant post of assistant manager. There are six candidates forming an initial universe of discourse $$ and have been scrutinized by recruitment committee. The committee further requires evaluation to select one of these candidates. The evaluation indicators are qualification $$, relevant experience in years $$, and computer skill $$. Their subparametric disjoint sets are $$, $$, and $$ respectively such that $$. Then, an ivfhs-set $$ is structured as $$, where

Definition 13. The complement of an ivfhs-set $$, denoted by $$, is defined by $$ where $$ and $$ stands for "not $$.

Example 2. Considering data from Example 1, we have

Its tabular representation is given in Table 2.

Definition 14. Let $$ be two ivfhs-sets then their hypersoft aggregation operations, i.e., $$ are given as:

• (1)

  Their union is an ivfhs-set defined by $$ such that $$ with maximum interval-valued fuzzy degrees respective to $$ and $$

• (2)

  Their intersection is an ivfhs-set defined by $$ such that $$ with minimum interval-valued fuzzy degrees respective to $$ and $$

Example 3. Considering Example 1, we have following two ivfhs-sets $$.

Definition 15. Let $$ be two ivfhs-sets, then their hypersoft product operations, i.e., $$ are given as:

• (1)

  The $$-product (AND-operation) is an ivfhs-set defined by $$ such that $$

• (2)

  The $$-product (OR-operation) is an ivfhs-set defined by $$ such that $$

Example 4. Considering the values of two ivfhs-sets $$ from Example 3, then

Their tabular representations are presented in Tables 3 and 4.

Definition 16. Let $$ be two ivfhs-sets then

• (1)

  $$ is said to be ivfhs $$-subset of $$, denoted by $$, if for every $$ such that $$

• (2)

  $$ and $$ are said to be ivfhs $$-equal, denoted by $$, if $$ and $$

Definition 17. Let $$ be two ivfhs-sets, then

• (1)

  $$ is said to be ivfs $$-subset of $$, denoted by $$, if for every $$ such that $$

• (2)

  $$ and $$ are said to be ivfs $$-equal, denoted by $$, if $$ and $$

Note: both $$ and $$ are named as ivfhs $$-inclusion and ivfhs $$-inclusion, respectively.

Proposition 18. If $$, then it implies $$.

Definition 19. Let $$ be two ivfhs-sets then $$ is said to be identical to $$, denoted by $$, if $$ and $$ for every $$.

Proposition 20. If $$, then it implies $$ which further implies $$.

Proof. Let $$, then by Definition 19, we have $$ and $$ for every $$. By Definition 17 part (1), we get $$. Now, applying Proposition 18, we obtain $$.

Proposition 21. The relations $$ and $$ satisfy the properties of equivalence relation on $$.

Proof. Applying the concept stated in Definitions 16 and Definitions 17, it is clear that both $$ and $$ satisfy reflexive property as $$ and $$. Their symmetric and transitive nature can also be deduced from these mentioned definitions. These properties collectively conclude that both $$ and $$ are equivalence relations.

We know from classical set theory, for a set D ≠ ∅ with a preorder ≤, an upward directed set is a set (D, ≤) in which every pair of elements in D has an upper bound, i.e., for d₁, d₂ ∈ D, there exists d₃ such that d₁ ≤ d₃ and d₂ ≤ d₃. The following definition is the generalized set theoretic version of upward directed set under hypersoft set environment.

Definition 22. An ivfhs-set $$ with $$ is called an upward directed ivfhs-set (UD-ivfhss) if for $$, there exists $$ such that

Example 5. Considering data from Example 1, we have ivfhs-set $$ as given below

Proposition 23. An ivfhs-set $$ with $$ is an UD-ivfhss if and only if $$ is an UD-ivfhss.

Proof. Let $$ be an UD-ivfhss then by definition of UD-ivfhss, then following conditions hold:

• (i)

  $$ and $$

• (ii)

  for $$ such that $$

The second condition implies that $$ and $$ which proves that $$ is an UD-ivfhss.

Conversely, let $$ is an UD-ivfhss, then the below given clauses hold due to definition of UD-ivfhss:

• (i)

  both $$ and $$ are nonempty sets

• (ii)

  for $$, there exists an upper bound for $$ and $$ in $$ which means $$ such that $$ and $$

The clause (ii) further implies $$ which shows that $$ is an UD-ivfhss.

Proposition 24. An ivfhs-set $$ with $$ is an UD-ivfhss if and only if $$.

Proof. Let $$ with $$ be an UD-ivfhss. Therefore, there exists $$ corresponding to pair $$ s.t.

Combining above equations, we obtain $$ which shows that $$ but we know that $$; hence, $$.

Conversely, let $$ then $$ implies $$, that is, there exists $$ corresponding to pair $$ s.t.

Corollary 25. Let $$ be an ivfhs-set with $$, then the given below statements are equivalent:

• (1)

  $$ is an UD-ivfhss over $$

• (2)

  $$ is an UD-ivfhss w.r.t ⊆

Proof. These can easily be verified by considering the consequences of Proposition 23 and Proposition 24.

Theorem 26. Let $$ be two ivfhs-sets, then

Theorem 27 (Generalized commutativity of ivfhs-sets). Let $$ be two ivfhs-sets, then

Theorem 28. Let $$ be three ivfhs-sets with $$, then

Proof.

• (1)

  Let $$ and $$. Since given that $$ which implies that there exists $$ for every $$ such that

Let $$, then by definition of $$, we have

By combining Equation (22) and Equation (23), we get

Similarly for $$, we get

From Equation (24) and Equation (25), we get $$, which shows that $$.

• (2)

  According to second part of Theorem 27, we have $$ which implies that

Therefore, from Equations (24), (25), and (26), it is vivid that $$ which shows that $$.

Part 3 and part 4 can easily be verified with the help of Theorem 27(2) and above results.

Theorem 29. Let $$ be two ivfhs-sets with $$ and $$, then

Theorem 30. Let $$ be three ivfhs-sets with $$ then

Proof.

• (1)

  Let $$ and $$. Since given that $$ which implies that there exists $$ for every $$ such that

Let $$, then by definition of $$, we have

By combining Equation (29) and Equation (30), we get

Similarly, for $$, we get

From Equation (31) and Equation (32), we get $$ which shows that $$.

• (2)

  According to Theorem 27, we have $$ which implies that

Therefore from Equations (31), (32), and (33), it is vivid that $$ which shows that $$.

Part 3 and part 4 can easily be verified with the help of Theorem 27(2) and above results.

Theorem 31. Let $$ be two ivfhs-sets with $$ and $$, then $$.

Theorem 32 (Generalized distributive inequalities of ivfhs-sets). Let $$ be three ivfhs-sets then

Theorem 33. Let $$ be two ivfhs-sets, then

Theorem 34. Let $$ be three ivfhs-sets with $$, then

Theorem 35. Let $$ be two ivfhs-sets with $$ and $$, then

Theorem 36. Let $$ be three ivfhs-sets with $$, then

Theorem 37. Let $$ be two ivfhs-sets with

Theorem 38. Let $$ be three ivfhs-sets, then $$.

Proof. From Theorem 32(1), we have

Corollary 39. Let $$ be three ivfhs-sets then

Proof. From Theorem 27(2), we have

Taking $$ on both sides of above inequality with $$, we have

Other parts can easily be validated in the similar manner.

Corollary 40. Let $$ be three ivfhs-sets then

Corollary 41. Let $$ be three ivfhs-sets, then

Corollary 42. Let $$ be three ivfhs-sets, then

Theorem 43. Let $$ be three ivfhs-sets. If $$, then

Proof. From Theorem 32(2), we have

Taking $$ on both sides of above inequality with $$, we have

Example 6. Let $$ be an initial universe and $$ be a set of attributes. The respective attribute-valued disjoint sets are $$, $$, $$, $$$$$$, and $$. Let us take $$, and $$ as subsets of $$, then we have three ivfhs-sets $$ with

It is clear that $$, $$, and $$; therefore, $$.

Consider $$ or $$ with

Now, let $$ or

Now, we find $$ with

Similarly, $$ with

Lastly, $$ with

It can be seen that

$$ is not valid in general.

Corollary 44. Let $$ be three ivfhs-sets. If $$, then

Proof. From Theorem 27(2), we have

Taking $$ on both sides of above inequality with $$, we have

Other parts can easily be validated in the similar manner.

Corollary 45. Let $$ be three ivfhs-sets. If $$, then

Corollary 46. Let $$ be three ivfhs-sets. If $$, then

Corollary 47. Let $$ be three ivfhs-sets. If $$, then

Theorem 48. Let $$ be three ivfhs-sets. If $$ and $$ is an UD-ivfhss, then we have

Proof. Since we know from Theorem 43 that

As given that $$ is an UD-ivfhss, therefore, $$ which implies $$ such that

Corollary 49. Let $$ be three ivfhs-sets. If $$ and $$ is an UD-ivfhss, then

Proof. Since we know from Theorem 27 that $$ which further implies that $$, i.e.,

By applying Theorem 36, we have

Hence,

Corollary 50. Let $$ be three ivfhs-sets. If $$ and $$ is an UD-ivfhss, then

Corollary 51. Let $$ be three ivfhs-sets. If $$ and $$ is an UD-ivfhss, then

Corollary 52. Let $$ be three ivfhs-sets. If $$ and $$ is an UD-ivfhss, then