A rough set approximates a subset using lower and upper bounds determined by equivalence classes, thereby capturing both certainty and uncertainty in membership [7, 9].

[Set] [10] A set is a well-defined collection of distinct elements or objects. If a is an element of a set A, we write a ∈ A; otherwise, we write a / ∈ A.

[Subset] [10] Let A and B be sets. A is called a subset of B, denoted A ⊆ B, if every element of A is also an element of B. If A ⊆ B but A ̸= B, then A is called a proper subset of B, denoted A ⊂ B.

[Empty Set] [10] The empty set, denoted by ∅, is the unique set containing no elements. Formally, for any set A, ∅ ⊆ A.

[Universal Set] A universal set, denoted by U, is the set that contains all elements under consideration in a particular context. Every set discussed is assumed to be a subset of U.

[Rough Set Approximation] [11] Let X be a nonempty universe of discourse, and let R ⊆ X ×X be an equivalence relation (also called an indiscernibility relation) on X. The relation R partitions X into disjoint equivalence classes, denoted by [x]R for each x ∈ X, where [x]R = {y ∈ X | (x, y) ∈ R}.

For any subset U ⊆ X, the lower approximation U and the upper approximation U are defined by:

(1) Lower Approximation:

U = {x ∈ X | [x]R ⊆ U}.

This set contains all elements whose entire equivalence class is contained within U; these elements definitely belong to U.

(2) Upper Approximation:

U = {x ∈ X | [x]R ∩ U ̸= ∅}.

This set contains all elements whose equivalence class has a nonempty intersection with U; these elements possibly belong to U.

Thus, the pair (U, U) forms the rough set representation of U, satisfying U ⊆ U ⊆ U .

We define Intuitionistic Rough Sets as follows [12, 13, 14].

[Intuitionistic Rough Set] (cf. [12, 14]) Let U be a nonempty universe and let R be an equivalence relation on U, which partitions U into equivalence classes {[x]R : x ∈ U}. For any subset X ⊆ U, define its lower approximation and upper approximation by X = x ∈ U [x]R ⊆ X , X = x ∈ U [x]R ∩ X ̸= ∅ .

An intuitionistic rough set corresponding to X is the ordered pair µX, νX , where µX : U −→ [0, 1] νX : U −→ [0, 1] are the membership and non-membership functions, respectively, satisfying 0 ≤ µX(x) + νX(x) ≤ 1 for every x ∈ U.

The hesitation margin (or uncertainty) at x is defined by πX(x) = 1 − µX(x) + νX(x) .

In the standard construction, µX and νX are assigned as follows:

(1) If x ∈ X, then x belongs entirely to X. Set µX(x) = 1, νX(x) = 0.

(2) If x / ∈ X, then the equivalence class [x]R does not intersect X. In this case, set µX(x) = 0, νX(x) = 1.

(3) If x ∈ X \ X (the boundary region), assign real values such that 0 < µX(x) < 1, 0 < νX(x) < 1, µX(x) + νX(x) < 1.

[Example of an Intuitionistic Rough Set] Consider the universe U = {a, b, c, d}, and suppose that an equivalence relation R partitions U into the following classes:

[a]R = {a, b}, [c]R = {c}, [d]R = {d}.

Let the set X ⊂ U be defined as X = {a, c}.

Then, the lower and upper approximations of X are:

X = { x ∈ U | [x]R ⊆ X } = {c},

X = { x ∈ U | [x]R ∩ X ̸= ∅ } = {a, b, c}.

Now, define the membership and non-membership functions µX and νX as follows:

• For x ∈ X (i.e., x = c):

µX(c) = 1, νX(c) = 0.

• For x / ∈ X (i.e., x = d):

µX(d) = 0, νX(d) = 1.

• For x ∈ X \ X (i.e., x = a and x = b, noting that a and b belong to the same equivalence class {a, b}):

choose µX(a) = µX(b) = 0.5, νX(a) = νX(b) = 0.3.

Then, the hesitation margin for these elements is πX(a) = πX(b) = 1 − (0.5 + 0.3) = 0.2.

Thus, the intuitionistic rough set corresponding to X is given by:

⟨µX, νX⟩ = {(a, 0.5, 0.3), (b, 0.5, 0.3), (c, 1, 0), (d, 0, 1)}.

We define One-directional S-rough Sets as follows [15, 16, 17].

[One-directional S-rough Set] [18] Let U be a non-empty universe and let R be an equivalence relation on U with equivalence classes denoted by [x] for each x ∈ U. Let F be a non-empty family of element transfer functions and fix some f ∈ F. For any subset X ⊂ U, define the f-extension of X by Xf = { u ∈ U \ X | f(u) ∈ X }.

Then the one-directional S-set of X is given by X = X ∪ Xf .

Next, define the lower approximation and upper approximation of X (with respect to the pair (R, F)) by (R, F)∗(X ) = { [x] | x ∈ U, [x] ⊆ X },

◦ ◦ ◦

(R, F) (X ) = { [x] | x ∈ U, [x] ∩ X ̸= ∅ }.

The ordered pair

◦ ◦ ◦

(R, F)∗(X ), (R, F) (X ) is called the one-directional S-rough set of X . Its boundary region is defined as

◦ ◦ ◦ ◦

BnR(X ) = (R, F) (X ) \ (R, F)∗(X ).

[A Concrete Example of a One-directional S-rough Set] Consider the universe U = {1, 2, 3, 4, 5}, with an equivalence relation R that partitions U into the equivalence classes [1] = {1, 2}, [3] = {3, 4}, [5] = {5}.

Let X = {1, 3}.

Define an element transfer function f (from a family F) by specifying f(2) = 1, f(4) is not defined or does not yield an element in X.

Then the f-extension of X is Xf = { u ∈ U \ X | f(u) ∈ X } = {2}, so that the one-directional S-set of X is X = X ∪ Xf = {1, 2, 3}.

Next, we compute the approximations with respect to R:

• The lower approximation is (R, F)∗(X ) = {[x] | [x] ⊆ {1, 2, 3}}.

Notice that the equivalence class [1] = {1, 2} is fully contained in X , whereas [3] = {3, 4} is not (since 4 ∈ / X ). Hence, (R, F)∗(X ) = [1] = {1, 2}.

• The upper approximation is (R, F) (X ) = {[x] | [x] ∩ {1, 2, 3} ̸= ∅}.

Both [1] and [3] have a non-empty intersection with X (since 1 ∈ [1] and 3 ∈ [3]); thus, (R, F) (X ) = [1] ∪ [3] = {1, 2} ∪ {3, 4} = {1, 2, 3, 4}.

Therefore, the one-directional S-rough set of X is given by

◦ ◦ ◦

(R, F)∗(X ), (R, F) (X ) = {1, 2}, {1, 2, 3, 4} , and its boundary region is BnR(X ) = {1, 2, 3, 4} \ {1, 2} = {3, 4}.

We define Tolerance Rough Sets as follows [19, 20, 21].

[Tolerance Rough Set] [22] Let U be a nonempty universe and let P be a set of attributes. For each attribute a ∈ P, define the similarity measure |a(x) − a(y)| SIMa(x, y) = 1 − , amax − amin where amax and amin denote the maximum and minimum values of a over U, respectively. Given a threshold τ ∈ [0, 1], we define the tolerance (or similarity) relation SIMP,τ on U by (x, y) ∈ SIMP,τ ⇐⇒ SIMa(x, y) ≥ τ.

For any x ∈ U, denote its tolerance class by SIMP,τ (x) = {y ∈ U | (x, y) ∈ SIMP,τ }.

Then, for any subset X ⊆ U the tolerance lower approximation of X is defined as Pτ X = {x ∈ U | SIMP,τ (x) ⊆ X}, and the tolerance upper approximation of X is defined as P X = {x ∈ U | SIMP,τ (x) ∩ X ̸= ∅}.

The ordered pair ⟨Pτ X, P X⟩ is called the tolerance rough set of X with respect to the tolerance relation SIMP,τ .

[A Concrete Example of a Tolerance Rough Set] Assume a universe U = {x1, x2, x3}, with a single real-valued attribute a (so that P = {a}). Suppose the attribute values are given by a(x1) = 1, a(x2) = 1.2, a(x3) = 2.

Then, amin = 1 and amax = 2. For any x, y ∈ U the similarity measure becomes SIMa(x, y) = 1 − |a(x) − a(y)|.

Thus, we obtain:

SIMa(x1, x2) = 1 − |1 − 1.2| = 0.8, SIMa(x1, x3) = 1 − |1 − 2| = 0, SIMa(x2, x3) = 1 − |1.2 − 2| = 0.2. Now, choose the threshold τ = 0.5. Then the tolerance relation SIM{a},τ is defined by (x, y) ∈ SIM{a},τ ⇐⇒ SIMa(x, y) ≥ 0.5.

Hence, the tolerance classes for each element are:

SIM{a},τ (x1) = {x1, x2}, SIM{a},τ (x2) = {x1, x2}, SIM{a},τ (x3) = {x3}.

Let X = {x1}. Then the tolerance approximations are computed as follows:

Pτ X = {x ∈ U | SIM{a},τ (x) ⊆ {x1}}.

Since for x1 we have SIM{a},τ (x1) = {x1, x2} ̸⊆ {x1} and similarly for x2 and x3, it follows that Pτ X = ∅.

On the other hand, the tolerance upper approximation is P X = {x ∈ U | SIM{a},τ (x) ∩ {x1} ̸= ∅}.

We have:

SIM{a},τ (x1) ∩ {x1} = {x1} ̸= ∅, SIM{a},τ (x2) ∩ {x1} = {x1} ̸= ∅, and SIM{a},τ (x3) ∩ {x1} = ∅.

P X = {x1, x2}.

The resulting tolerance rough set of X is therefore given by ⟨Pτ X, P X⟩ = ⟨∅, {x1, x2}⟩.

We define Dynamic Rough Sets as follows [23, 24, 25].

[Dynamic Rough Set] [23] Let A = (U, P) be an information system where • U is a nonempty finite set of objects, and • P is a set of attributes.

Let X ⊆ U be the target (concept) set and let T ⊆ P be a subset of attributes chosen as dynamic criteria. Moreover, let d and d be fixed threshold constants (called, respectively, the inward transfer standard and the outward transfer standard).

For each object x ∈ U, we define two coefficients:

ρ (x) = (inward transfer coefficient) ρ (x) = (outward transfer coefficient) which quantitatively measure, with respect to the dynamic criteria T, the potential for an object currently not in X to be included, and for an object in X to be excluded, respectively. (In practical applications these coefficients are often computed by aggregating measures on each attribute in T or, in many cases, are simply chosen as constant values for simplicity.) Then we define:

(1) The inflated dynamic main set (or candidate addition set) IT (X) = {x ∈ U \ X : ρ (x) ≥ d }.

(2) The contracted dynamic set (or candidate removal set) CT (X) = {x ∈ X : ρ (x) ≥ d }.

(3) The resulting two-direction dynamic set is defined as DT (X) = X ∪ IT (X) \ CT (X).

Finally, if we take a subset Q ⊆ P of attributes and use it to induce an equivalence relation on U (with equivalence classes denoted by [x]Q), the dynamic rough approximations of X are given by DT (X) = {x ∈ U : [x]Q ⊆ DT (X)} (dynamic lower approximation) DT (X) = {x ∈ U : [x]Q ∩ DT (X) ̸= ∅} (dynamic upper approximation).

Consider an information system A = (U, P) with U = {u1, u2, u3, u4} and P = {p1, p2}.

Assume that the initial target set is X = {u1, u2}, and let the dynamic criteria be T = {p1}.

Suppose we define the transfer coefficients as follows:

0.8, if x has p1 valued “high”, 0.6, if x has p1 valued “low”, ρ (x) = ρ (x) = 0.4, otherwise, 0.2, otherwise.

Let the threshold constants be d = 0.7 and d = 0.5.

Assume that:

• In X, the object u1 has p1 = “high” so that ρ (u1) = 0.2, and u2 has p1 = “low” so that ρ (u2) = 0.6. • In U \ X, the objects u3 and u4 have p1 = “high” so that ρ (u3) = ρ (u4) = 0.8.

Then we have:

IT (X) = {u3, u4} since 0.8 ≥ 0.7, CT (X) = {u2} since 0.6 ≥ 0.5.

Thus, the two-direction dynamic set is computed as DT (X) = (X ∪ IT (X)) \ CT (X) = {u1, u2} ∪ {u3, u4} \ {u2} = {u1, u3, u4}.

Now, suppose that the attribute set Q = P induces the following partition on U:

[u1]Q = [u3]Q = {u1, u3} and [u2]Q = [u4]Q = {u2, u4}.

Then the dynamic lower approximation is DT (X) = {x ∈ U : [x]Q ⊆ DT (X)} = {u1, u3}, because for every x ∈ {u1, u3}, the entire equivalence class [x]Q = {u1, u3} is contained in DT (X), while for x ∈ {u2, u4} the equivalence class {u2, u4} is not fully contained in DT (X) (since u2 ∈ / DT (X)).

Similarly, the dynamic upper approximation is DT (X) = {x ∈ U : [x]Q ∩ DT (X) ̸= ∅} = U, since every equivalence class (whether {u1, u3} or {u2, u4}) has a non-empty intersection with DT (X).

This example illustrates how the dynamic rough set extends the classical rough set by allowing the expansion (through IT (X)) and contraction (through CT (X)) of the set X based on dynamic criteria.

The HyperRough Set extends rough set theory by incorporating multiple attributes. Its formal definition is given below [26, 27, 28, 29].

[HyperRough Set] [26, 27] Let X be a nonempty finite universe, and let T1, T2, . . . , Tn be n distinct attributes with corresponding domains J1, J2, . . . , Jn. Define the Cartesian product J = J1 × J2 × · · · × Jn.

Let R ⊆ X × X be an equivalence relation on X, with [x]R denoting the equivalence class of x. A HyperRough Set over X is a pair (F, J), where:

• F : J → P(X) is a mapping that assigns to each attribute value combination a = (a1, a2, . . . , an) ∈ J a subset F(a) ⊆ X.

• For each a ∈ J, the rough set approximations of F(a) are defined as F(a) = {x ∈ X | [x]R ⊆ F(a)}, F(a) = {x ∈ X | [x]R ∩ F(a) ̸= ∅}.

Here, F(a) comprises all elements whose equivalence classes are completely contained within F(a), while F(a) contains elements whose equivalence classes intersect F(a). Additionally, the following properties hold for all a ∈ J:

• F(a) ⊆ F(a).

• If F(a) = ∅, then F(a) = F(a) = ∅.

• If F(a) = X, then F(a) = F(a) = X.

[HyperRough Set] Let X = {u1, u2, u3, u4, u5, u6} be a nonempty finite universe. Consider two attributes:

T1 : Color with domain J1 = {Red, Blue}, T2 : Size with domain J2 = {Small, Large}.

The Cartesian product of the attribute domains is J = J1 × J2 = {(Red, Small), (Red, Large), (Blue, Small), (Blue, Large)}.

Define the mapping F : J → P(X) by F(Red, Small) = {u1, u2}, F(Red, Large) = {u3, u4}, F(Blue, Small) = {u2, u5}, F(Blue, Large) = {u4, u6}.

Assume that an equivalence relation R ⊆ X × X is defined by the partition [u1]R = {u1, u2}, [u3]R = {u3, u4}, [u5]R = {u5, u6}.

For each a ∈ J the rough set approximations are defined as follows:

F(a) = {x ∈ X | [x]R ⊆ F(a)}, F(a) = {x ∈ X | [x]R ∩ F(a) ̸= ∅}.

Computation for a = (Red, Small):

Since F(Red, Small) = {u1, u2}, consider the equivalence class for any x ∈ X:

• For x = u1 (or u2), we have [u1]R = {u1, u2} ⊆ {u1, u2}. Thus, u1, u2 ∈ F(Red, Small).

• For any x with x ∈ {u3, u4} or x ∈ {u5, u6}, the entire equivalence class is not a subset of F(Red, Small). Therefore, F(Red, Small) = {u1, u2}.

For the upper approximation, note that • The equivalence class [u1]R = {u1, u2} intersects F(Red, Small) (in fact, it equals F(Red, Small)). • The classes {u3, u4} and {u5, u6} have no intersection with F(Red, Small).

F(Red, Small) = {u1, u2}.

Computation for a = (Blue, Small):

Here, F(Blue, Small) = {u2, u5}.

Now, consider the equivalence classes:

• For x in [u1]R = {u1, u2}: Since u2 ∈ F(Blue, Small) but u1 ∈ / F(Blue, Small), we have [u1]R ⊈ F(Blue, Small). Thus, {u1, u2} ̸⊆ F(Blue, Small).

• For x in [u5]R = {u5, u6}: u5 ∈ F(Blue, Small) but u6 ∈ / F(Blue, Small). Hence, [u5]R ⊈ F(Blue, Small). Therefore, the lower approximation is empty:

F(Blue, Small) = ∅.

For the upper approximation, include every x ∈ X with [x]R ∩ F(Blue, Small) ̸= ∅:

• The class [u1]R = {u1, u2} intersects F(Blue, Small) (via u2).

• The class [u5]R = {u5, u6} intersects F(Blue, Small) (via u5).

F(Blue, Small) = {u1, u2, u5, u6}.

This example concretely demonstrates a HyperRough Set with explicit computation of the lower and upper approximations for different attribute combinations.

An n-SuperHyperRough Set generalizes rough sets by using power sets of attribute values to produce nuanced approximations under uncertainty[26, 30, 27]. The definition of n-SuperHyperRough Sets is described as follows. [n-SuperHyperRough Set] [26, 27] Let X be a nonempty finite universe, and let T1, T2, . . . , Tn be n distinct attributes with respective domains J1, J2, . . . , Jn. For each attribute Ti, let P(Ji) denote its power set. Define the set of all possible attribute value combinations as J = P(J1) × P(J2) × · · · × P(Jn).

Let R ⊆ X × X be an equivalence relation on X. An n-SuperHyperRough Set over X is a pair (F, J), where:

• F : J → P(X) is a mapping that assigns to each attribute value combination A = (A1, A2, . . . , An) ∈ J (with Ai ⊆ Ji for all i) a subset F(A) ⊆ X.

• For each A ∈ J, the lower and upper approximations are defined as F(A) = {x ∈ X | [x]R ⊆ F(A)}, F(A) = {x ∈ X | [x]R ∩ F(A) ̸= ∅}.

Thus, F(A) consists of all elements whose equivalence classes are entirely contained in F(A), and F(A) includes those elements whose equivalence classes intersect F(A). The following properties hold for all A ∈ J:

• F(A) ⊆ F(A).

• If F(A) = ∅, then F(A) = F(A) = ∅.

• If F(A) = X, then F(A) = F(A) = X.

• For any A, B ∈ J, F(A ∩ B) ⊆ F(A) ∩ F(B), F(A ∪ B) ⊇ F(A) ∪ F(B).

[n-SuperHyperRough Set] Let X = {u1, u2, u3, u4} be a nonempty finite universe. Consider two attributes:

T1 : Shape with domain J1 = {Circle, Square}, T2 : Color with domain J2 = {Red, Blue}.

For each attribute, consider its power set:

P(J1) = {∅, {Circle}, {Square}, {Circle, Square}}, P(J2) = {∅, {Red}, {Blue}, {Red, Blue}}.

The set of all possible attribute value combinations is J = P(J1) × P(J2).

Define a mapping F : J → P(X) by specifying values on selected elements of J. For instance, let F({Circle}, {Red}) = {u1, u2}, F({Square}, {Blue}) = {u3}, F({Circle, Square}, {Red, Blue}) = X, F(A) = ∅ for all other A ∈ J.

Assume an equivalence relation R ⊆ X × X defined by the partition [u1]R = {u1, u2}, [u3]R = {u3, u4}.

For any A ∈ J the rough approximations are defined as:

F(A) = {x ∈ X | [x]R ⊆ F(A)}, F(A) = {x ∈ X | [x]R ∩ F(A) ̸= ∅}.

Computation for A = ({Circle}, {Red}):

Since F({Circle}, {Red}) = {u1, u2}, we have:

[leftmargin=2em]For x = u1 (or u2), [u1]R = {u1, u2} ⊆ {u1, u2}. Thus, u1, u2 ∈ F({Circle}, {Red}). No other equivalence class is completely contained in F({Circle}, {Red}).

Therefore, F({Circle}, {Red}) = {u1, u2}.

For the upper approximation, note:

[leftmargin=2em]The equivalence class [u1]R = {u1, u2} intersects F({Circle}, {Red}).

F({Circle}, {Red}) = {u1, u2}.

Computation for A = ({Square}, {Blue}):

Here, F({Square}, {Blue}) = {u3}.

Now, consider the equivalence class:

[leftmargin=2em]For x = u3 or x = u4, the equivalence class [u3]R = {u3, u4} is not completely contained in {u3} (since u4 ∈ { / u3}). Hence, F({Square}, {Blue}) = ∅.

However, since u3 ∈ F({Square}, {Blue}) and [u3]R intersects F({Square}, {Blue}), F({Square}, {Blue}) = [u3]R = {u3, u4}.

Computation for A = ({Circle, Square}, {Red, Blue}):

Since F({Circle, Square}, {Red, Blue}) = X, we immediately have F(A) = {x ∈ X | [x]R ⊆ X} = X, F(A) = {x ∈ X | [x]R ∩ X ̸= ∅} = X.

This example illustrates an n-SuperHyperRough Set where attribute values are chosen from the power sets of the original domains. The mapping F is defined on J = P(J1) × P(J2) and, together with the equivalence relation R, yields concrete lower and upper approximations.

Let X be a nonempty finite universe and let J = J1 × J2 × · · · × Jn be the set of all attribute value combinations corresponding to the attributes T1, . . . , Tn. Let R be an equivalence relation on X. An Intuitionistic Hyperrough Set over X is a pair (F, J) where F : J → I(X) is a mapping that assigns to each attribute combination a = (a1, a2, . . . , an) ∈ J an intuitionistic set F(a) = ⟨x, µF (a)(x), νF (a)(x)⟩ | x ∈ X with the property that for each a ∈ J we define the intuitionistic lower and upper approximations by:

F(a) = x ∈ X | ∀ y ∈ [x]R, µF (a)(y) = 1 and νF (a)(y) = 0 ,

F(a) = x ∈ X | [x]R ∩ {y ∈ X | µF (a)(y) > 0} ̸= ∅ .

In this way, the pair F(a), F(a) serves as the intuitionistic rough approximation of the intuitionistic set F(a). Notice that by construction, F(a) ⊆ F(a), and if for each x we have the crisp condition µF (a)(x) ∈ {0, 1} (and hence νF (a)(x) = 1 − µF (a)(x)), the pair of approximations coincide with the classical rough approximations of F(a).

The following theorem establishes that Intuitionistic Hyperrough Sets generalize both Hyperrough Sets and Intuitionistic Rough Sets.

[Generalization by Intuitionistic Hyperrough Sets] Let X, J, R, and F be as in Definition . Then:

(i) • If, for every a ∈ J, the mapping F(a) is crisp (that is, for every x ∈ X, µF (a)(x) ∈ {0, 1} and νF (a)(x) = 1 − µF (a)(x)), then the Intuitionistic Hyperrough Set (F, J) coincides with the classical Hyperrough Set.

(ii) If n = 1 (i.e. J reduces to J1), then (F, J) becomes an Intuitionistic Rough Set.

Proof : (i) Suppose that for every a ∈ J and x ∈ X we have µF (a)(x) ∈ {0, 1} and νF (a)(x) = 1 − µF (a)(x). Define F (a) = {x ∈ X | µF (a)(x) = 1}.

Then the intuitionistic lower approximation F(a) = {x ∈ X | ∀ y ∈ [x]R, µF (a)(y) = 1} is exactly the same as the classical lower approximation F (a) = {x ∈ X | [x]R ⊆ F (a)}, and similarly for the upper approximation. Hence, the pair F(a), F(a) recovers the classical rough approximations assigned by the Hyperrough Set (F , J). Thus, Intuitionistic Hyperrough Sets generalize Hyperrough Sets.

(ii) If n = 1, then J = J1 and the mapping reduces to F : J1 → I(X).

In this case, for each attribute value a ∈ J1, the pair F(a), F(a) defines the intuitionistic rough approximations of the intuitionistic set F(a). This exactly matches the standard definition of an Intuitionistic Rough Set defined on X. Hence, the model reduces appropriately. □

We define Intuitionistic Superhyperrough Sets as follows.

[Intuitionistic Superhyperrough Set] Let X be a nonempty finite universe and let J = P(J1) × P(J2) × · · · × P(Jn) be the Cartesian product of the power sets of the attribute domains Ji. Let R be an equivalence relation on X. An Intuitionistic Superhyperrough Set over X is a pair (G, J ) where G : J → I(X) is a mapping that assigns to each attribute set combination A = (A1, A2, . . . , An) ∈ J (Ai ⊆ Ji) an intuitionistic set n o G(A) = ⟨x, µG(A)(x), νG(A)(x)⟩ | x ∈ X .

For each A ∈ J , define the intuitionistic lower and upper approximations by G(A) = x ∈ X | ∀ y ∈ [x]R, µG(A)(y) = 1 and νG(A)(y) = 0 ,

G(A) = x ∈ X | [x]R ∩ {y ∈ X | µG(A)(y) > 0} ̸= ∅ .

Thus, the pair G(A), G(A) is the intuitionistic rough approximation of the intuitionistic set G(A).

The following theorem shows that Intuitionistic Superhyperrough Sets generalize both SuperHyperrough Sets and Intuitionistic Hyperrough Sets.

[Generalization by Intuitionistic Superhyperrough Sets] Let X, J , R, and G be as in Definition . Then:

(i) If for every A ∈ J the mapping G(A) is crisp (that is, for every x ∈ X, µG(A)(x) ∈ {0, 1} and νG(A)(x) = 1 − µG(A)(x)), then (G, J ) coincides with a classical n-SuperHyperrough Set.

(ii) If the mapping G is restricted to the case where each component Ai is a singleton (so that J is effectively reduced to the Cartesian product J1 × · · · × Jn), then (G, J ) reduces to an Intuitionistic Hyperrough Set.

Proof : (i) Assume that for all A ∈ J and for all x ∈ X the membership and nonmembership functions satisfy µG(A)(x) ∈ {0, 1} and νG(A)(x) = 1 − µG(A)(x). Define G (A) = {x ∈ X | µG(A)(x) = 1}.

Then the intuitionistic approximations G(A) = {x ∈ X | [x]R ⊆ G (A)} and G(A) = {x ∈ X | [x]R ∩ G (A) ̸= ∅} coincide with the lower and upper approximations of a classical SuperHyperrough Set (G , J ).

(ii) Now suppose that G is restricted to those A = (A1, A2, . . . , An) ∈ J for which each Ai is a singleton, i.e., Ai = {ai} for some ai ∈ Ji. In this case, the set J is isomorphic to the Cartesian product J = J1 × J2 × · · · × Jn.

Then the mapping G restricted to this subdomain is exactly a mapping F : J → I(X), as in Definition . Therefore, the approximations G(A) and G(A) coincide with those defined by F, and the model reduces to an Intuitionistic Hyperrough Set. This shows that Intuitionistic Superhyperrough Sets generalize Intuitionistic Hyperrough Sets. □

We define One–directional S–Hyperrough Set as follows.

[One–directional S–Hyperrough Set] Let X, R and J be as above. Suppose a mapping F : J → P(X) is given. For each a ∈ J, define its one–directional S–extension by F(a) = F(a) ∪ { u ∈ X \ F(a) | f(u) ∈ F(a) }.

Then, the lower and upper approximations of F(a) are defined as F (a) = {x ∈ X | [x]R ⊆ F(a) }, F (a) = {x ∈ X | [x]R ∩ F(a) ̸= ∅}.

The pair F (a), F (a) is the intuitionistic rough approximation of F(a) for each a ∈ J. The collection { (F (a), F (a)) : a ∈ J } is called a One–directional S–Hyperrough Set over X.

[One–directional S–Hyperrough Set] Let X = {1, 2, 3, 4, 5}, and let the equivalence relation R partition X as follows:

[1]R = {1, 2}, [3]R = {3, 4}, [5]R = {5}.

Suppose the attribute domain is J = {A, B}, and define the mapping F : J → P(X) by F(A) = {1, 3, 5} and F(B) = {2, 4}.

Let the element transfer function f : X → X be given by f(2) = 1, f(4) = 3, and f(x) = x for x ∈ {1, 3, 5}.

Then the one–directional S–extension is defined by F(a) = F(a) ∪ { u ∈ X \ F(a) | f(u) ∈ F(a) } for a ∈ J.

For a = A:

F(A) = {1, 3, 5}.

The elements not in F(A) are X \ F(A) = {2, 4}. Since f(2) = 1 ∈ F(A) and f(4) = 3 ∈ F(A), we have F(A) = {1, 3, 5} ∪ {2, 4} = X.

For a = B:

F(B) = {2, 4}.

Now, X \ F(B) = {1, 3, 5} and for each u ∈ {1, 3, 5} we find f(1) = 1 ∈ { / 2, 4}, f(3) = 3 ∈ { / 2, 4}, f(5) = 5 ∈ { / 2, 4}.

F(B) = F(B) = {2, 4}.

Next, we form the rough approximations using the equivalence relation R.

For a = A: Since F(A) = X, every equivalence class is contained in X; hence, F (A) = {x ∈ X | [x]R ⊆ X} = X, and F (A) = {x ∈ X | [x]R ∩ X ̸= ∅} = X.

For a = B: We have F(B) = {2, 4}. Then the lower approximation is F (B) = {x ∈ X | [x]R ⊆ {2, 4}}.

Notice:

[1]R = {1, 2} ̸⊆ {2, 4}, [3]R = {3, 4} ̸⊆ {2, 4}, [5]R = {5} ̸⊆ {2, 4}.

F (B) = ∅.

The upper approximation is F (B) = {x ∈ X | [x]R ∩ {2, 4} ̸= ∅}.

Here, [1]R ∩ {2, 4} = {2} ̸= ∅, [3]R ∩ {2, 4} = {4} ̸= ∅, [5]R ∩ {2, 4} = ∅.

So, F (B) = {1, 2, 3, 4}.

Thus, the One–directional S–Hyperrough Set is given by:

◦ ◦ ◦ ◦

{(F (A), F (A)), (F (B), F (B))} = {(X, X), (∅, {1, 2, 3, 4})}.

[Generalization Property of One–directional S–Hyperrough Sets] Let F : J → P(X) be as in Definition .

(i) If the transfer function f is trivial (i.e., if f(u) = u for all u ∈ X, so that F(a) = F(a) for all a ∈ J), then the One–directional S–Hyperrough Set { (F (a), F (a)) : a ∈ J } coincides with the classical Hyperrough Set defined by F.

(ii) If n = 1 so that J = J1, then the model reduces to the one–directional S–rough set.

Proof : (i) Assume that f is trivial, that is, f(u) = u for every u ∈ X. Then for any F(a) ⊆ X we have F(a) = F(a) ∪ {u ∈ X \ F(a) | f(u) ∈ F(a)} = F(a) ∪ {u ∈ X \ F(a) | u ∈ F(a)} = F(a).

Consequently, the approximations become F (a) = {x ∈ X | [x]R ⊆ F(a)} and F (a) = {x ∈ X | [x]R ∩ F(a) ̸= ∅}, which are exactly the classical rough set approximations for F(a). Hence, the One–directional S–Hyperrough Set reduces to the classical Hyperrough Set.

(ii) If n = 1, then J is simply J1 and the mapping F : J1 → P(X) defines for each attribute value a ∈ J1 a subset F(a) ⊆ X. The one–directional S–extension F(a) is then used to form the lower and upper approximations exactly as in the definition of a one–directional S–rough set. Thus, the One–directional S–Hyperrough Set (which in this case is indexed by the single attribute domain J1) coincides with the one–directional S–rough set. □

We define One–directional S–SuperHyperrough Set as follows.

[One–directional S–Superhyperrough Set] Let X be a nonempty finite universe, and let J = P(J1) × P(J2) × · · · × P(Jn) be the Cartesian product of the power sets of the attribute domains. Suppose that a mapping G : J → P(X) is given. For each A = (A1, A2, . . . , An) ∈ J , define its one–directional S–extension by G(A) = G(A) ∪ { u ∈ X \ G(A) | f(u) ∈ G(A) }.

Then the lower and upper approximations of G(A) are defined as G (A) = {x ∈ X | [x]R ⊆ G(A) }, G (A) = {x ∈ X | [x]R ∩ G(A) ̸= ∅}.

The collection

◦ ◦ ∗

{ (G (A), G (A)) : A ∈ J } is called a One–directional S–Superhyperrough Set over X.

[One–directional S–Superhyperrough Set] Let X = {1, 2, 3, 4}, and let the equivalence relation R partition X as [1]R = {1, 2} and [3]R = {3, 4}.

Consider a single attribute T1 with domain J1 = {red, blue}.

Then, the power set is J = P(J1) = {∅, {red}, {blue}, {red, blue}}.

Define the mapping G : J → P(X) by G(∅) = ∅, G({red}) = {1, 3}, G({blue}) = {2, 4}, G({red, blue}) = X.

Let the transfer function f : X → X be given by f(2) = 1, f(4) = 3, f(x) = x for x ∈ {1, 3}.

Then, for each A ∈ J the one–directional S–extension is defined by G(A) = G(A) ∪ { u ∈ X \ G(A) | f(u) ∈ G(A)}.

We compute each case:

Case 1. A = {red}: G({red}) = {1, 3}. Then, X \ G({red}) = {2, 4}.

Since f(2) = 1 ∈ {1, 3} and f(4) = 3 ∈ {1, 3}, we obtain G({red}) = {1, 3} ∪ {2, 4} = X.

Thus, the rough approximations are:

G ({red}) = {x ∈ X | [x]R ⊆ X} = X, G ({red}) = X.

Case 2. A = {blue}: G({blue}) = {2, 4}. Then, X \ G({blue}) = {1, 3}.

For u ∈ {1, 3}, we have f(1) = 1 ∈ { / 2, 4} and f(3) = 3 ∈ { / 2, 4}. Hence, G({blue}) = {2, 4}.

The lower approximation is G ({blue}) = {x ∈ X | [x]R ⊆ {2, 4}}.

Since [1]R = {1, 2} ̸⊆ {2, 4} and [3]R = {3, 4} ̸⊆ {2, 4}, it follows that G ({blue}) = ∅.

The upper approximation is G ({blue}) = {x ∈ X | [x]R ∩ {2, 4} ̸= ∅}.

We have:

[1]R ∩ {2, 4} = {2} ̸= ∅, [3]R ∩ {2, 4} = {4} ̸= ∅.

G ({blue}) = {1, 2, 3, 4} = X.

Case 3. A = {red, blue}: By definition, G({red, blue}) = X, so that G({red, blue}) = X.

Thus, both approximations equal X.

Case 4. A = ∅: Here, G(∅) = ∅. Then clearly, G(∅) = ∅, and the lower and upper approximations are both empty.

Summarizing, the One–directional S–Superhyperrough Set over X is the collection

◦ ◦ ∗

{ (G (A), G (A)) : A ∈ J }, with A = ∅ : (∅, ∅), A = {red} : (X, X), A = {blue} : (∅, X), A = {red, blue} : (X, X).

[Generalization Property of One–directional S–Superhyperrough Sets] Let G : J → P(X) be as in Definition .

(i) If, for every A ∈ J , the mapping G(A) is crisp (that is, if for every x ∈ X we have x ∈ G(A) or x / ∈ G(A) so that the one–directional extension is trivial), then the One–directional S–Superhyperrough Set coincides with the classical Superhyperrough Set.

(ii) If the mapping G is restricted to those A = (A1, . . . , An) ∈ J where every Ai is a singleton (so that J is isomorphic to J = J1 × · · · × Jn), then the One–directional S–Superhyperrough Set reduces to a One–directional S–Hyperrough Set.

Proof : (i) Suppose that for every A ∈ J the set G(A) is crisp, i.e., for each x ∈ X there is no uncertainty (the one–directional S–extension is redundant). Then G(A) = G(A) ∪ {u ∈ X \ G(A) | f(u) ∈ G(A)} = G(A).

Thus, the approximations become G (A) = {x ∈ X | [x]R ⊆ G(A)} and G (A) = {x ∈ X | [x]R ∩ G(A) ̸= ∅}.

These are exactly the classical definitions for an n-Superhyperrough Set. Therefore, the One–directional S–Superhyperrough Set generalizes the classical Superhyperrough Set.

(ii) If n = 1 then J = J1 and the mapping F : J1 → P(X) assigns to every single attribute value a subset F(a) ⊆ X. The tolerance approximations defined by Fτ (a) = {x ∈ X | SIMP,τ (x) ⊆ F(a)} and F (a) = {x ∈ X | SIMP,τ (x) ∩ F(a) ̸= ∅} are exactly those given by the standard tolerance rough set model. Hence, the Tolerance Hyperrough Set reduces to a Tolerance Rough Set. □

[Tolerance Hyperrough Set] Let X = {1, 2, 3, 4, 5}.

Consider two condition attributes with domains J1 = {Red, Blue} and J2 = {High, Low}.

Then, the Cartesian product is J = J1 × J2 = {(Red, High), (Red, Low), (Blue, High), (Blue, Low)}.

Define the mapping F : J → P(X) by F(Red, High) = {1, 2, 3}, F(Red, Low) = {3, 4, 5}, and for simplicity we let F(Blue, High) = {1, 2, 3}, F(Blue, Low) = {3, 4, 5}.

Assume a tolerance relation on X specified as follows. For each x ∈ X, define {1, 2} if x = 1, {1, 2, 3} if x = 2, SIM(x) = {2, 3, 4} if x = 3, {3, 4, 5} if x = 4, {4, 5} if x = 5.

Then, for any subset S ⊆ X, Pτ S = {x ∈ X | SIM(x) ⊆ S} and P S = {x ∈ X | SIM(x) ∩ S ̸= ∅}.

Case: Consider the attribute combination (Red, High). Then F(Red, High) = {1, 2, 3}.

Lower approximation:

• For x = 1: SIM(1) = {1, 2} ⊆ {1, 2, 3} so 1 ∈ Pτ F(Red, High).

• For x = 2: SIM(2) = {1, 2, 3} ⊆ {1, 2, 3} so 2 ∈ Pτ F(Red, High).

• For x = 3: SIM(3) = {2, 3, 4} but 4 ∈ { / 1, 2, 3}; hence 3 ∈ / Pτ F(Red, High).

• x = 4 and x = 5 are not in F(Red, High) so we do not include them.

Pτ F(Red, High) = {1, 2}.

Upper approximation:

• For x = 1: SIM(1) = {1, 2} intersects {1, 2, 3} (in fact, {1, 2} ̸= ∅); so 1 ∈ P F(Red, High).

• For x = 2: SIM(2) = {1, 2, 3} clearly intersects {1, 2, 3}; so 2 ∈ P F(Red, High).

• For x = 3: SIM(3) = {2, 3, 4} intersects {1, 2, 3} (since 2, 3 ∈ {1, 2, 3}); so 3 ∈ P F(Red, High). • For x = 4: SIM(4) = {3, 4, 5} intersects {1, 2, 3} (since 3 ∈ {1, 2, 3}); so 4 ∈ P F(Red, High).

• For x = 5: SIM(5) = {4, 5} does not intersect {1, 2, 3}; so 5 ∈ / P F(Red, High).

P F(Red, High) = {1, 2, 3, 4}.

Therefore, the Tolerance Hyperrough Set corresponding to the attribute combination (Red, High) is the pair Pτ F(Red, High), P F(Red, High) = ({1, 2}, {1, 2, 3, 4}).

Classically, a superhyperrough set is defined as the collection of maximal elements (with respect to set inclusion) of a hyperrough set. We generalize this notion as follows.

[Tolerance Superhyperrough Set] Let X be a nonempty finite universe, and let J = P(J1) × P(J2) × · · · × P(Jn) be the Cartesian product of the power sets of the attribute domains. Let G : J → P(X) be a mapping. For each A = (A1, . . . , An) ∈ J , define the tolerance approximations by Gτ (A) := {x ∈ X | SIMP,τ (x) ⊆ G(A)}, G (A) := {x ∈ X | SIMP,τ (x) ∩ G(A) ̸= ∅}.

Then the pair Gτ (A), G (A) represents the tolerance rough approximation of G(A). The family { (Gτ (A), G (A)) | A ∈ J } is called a Tolerance Superhyperrough Set over X.

[Generalization of Superhyperrough and Tolerance Hyperrough Sets] Let G : J → P(X) be as in Definition . Then:

(i) If for every A ∈ J the mapping G(A) is crisp (that is, the tolerance approximations yield no uncertainty), then the tolerance approximations Gτ (A) and G (A) coincide with the classical rough approximations. In this case, the Tolerance Superhyperrough Set is equivalent to the classical Superhyperrough Set. (ii) If G is restricted to those elements A = (A1, . . . , An) ∈ J where each Ai is a singleton (so that J is isomorphic to J = J1 × · · · × Jn), then the tolerance approximations reduce to those given in Definition ; that is, the Tolerance Superhyperrough Set reduces to a Tolerance Hyperrough Set.

Proof : (i) If G(A) is crisp for every A ∈ J then for every x ∈ X we have that x ∈ G(A) or x / ∈ G(A). In this situation, the tolerance approximations simplify to Gτ (A) = {x ∈ X | [x]SIM ⊆ G(A)}, and G (A) = {x ∈ X | [x]SIM ∩ G(A) ̸= ∅}, which are exactly those of the classical n-Superhyperrough Set. (In essence, when there is no additional uncertainty induced by the tolerance relation, the model is classical.)

(ii) Suppose now that G is restricted to those A = (A1, . . . , An) ∈ J in which every Ai is a singleton. Then J is naturally isomorphic to J = J1 × J2 × · · · × Jn.

Define a mapping F : J → P(X) by F(a1, . . . , an) = G({a1}, . . . , {an}).

Then for every a ∈ J the tolerance approximations for F(a) are precisely those defined in Definition . Thus, under this restriction, the Tolerance Superhyperrough Set reduces to a Tolerance Hyperrough Set. □ [Tolerance Superhyperrough Set] Let X = {1, 2, 3, 4}.

Consider a single attribute T1 with domain J1 = {red, blue}.

Then, the power set of the domain is P(J1) = {∅, {red}, {blue}, {red, blue}}.

Define a mapping G : P(J1) → P(X) by G(∅) = ∅, G({red}) = {1, 3}, G({blue}) = {2, 4}, G({red, blue}) = X.

Assume a tolerance relation on X given by:

SIM(1) = {1, 2}, SIM(2) = {1, 2, 3}, SIM(3) = {2, 3, 4}, SIM(4) = {3, 4}.

Then for any S ⊆ X, Pτ S = {x ∈ X | SIM(x) ⊆ S} and P S = {x ∈ X | SIM(x) ∩ S ̸= ∅}.

We now compute the tolerance approximations for several cases.

Case 1. A = ∅. Then, G(∅) = ∅. Hence, Pτ G(∅) = ∅ and P G(∅) = ∅.

Case 2. A = {red}. Then, G({red}) = {1, 3}.

Lower approximation:

Check each x ∈ X:

• For x = 1: SIM(1) = {1, 2}. Since {1, 2} ̸⊆ {1, 3} (because 2 ∈ { / 1, 3}), 1 ∈ / Pτ G({red}).

• For x = 2: SIM(2) = {1, 2, 3} is not a subset of {1, 3} (as 2 ∈ { / 1, 3}).

• For x = 3: SIM(3) = {2, 3, 4} is not contained in {1, 3} (since 2 and 4 are missing).

• For x = 4: SIM(4) = {3, 4} is not contained in {1, 3} (since 4 ∈ { / 1, 3}).

Pτ G({red}) = ∅.

Upper approximation:

• x = 1: SIM(1) = {1, 2} and {1, 2} ∩ {1, 3} = {1} ̸= ∅; so 1 ∈ P G({red}).

• x = 2: SIM(2) = {1, 2, 3} intersects {1, 3} (since both 1 and 3 are in the intersection); so 2 ∈ P G({red}). • x = 3: SIM(3) = {2, 3, 4} intersects {1, 3} (via element 3); so 3 ∈ P G({red}).

• x = 4: SIM(4) = {3, 4} intersects {1, 3} (since 3 ∈ {1, 3}); so 4 ∈ P G({red}).

P G({red}) = {1, 2, 3, 4}.

Case 3. A = {red, blue}. Then, G({red, blue}) = X = {1, 2, 3, 4}. Hence, both approximations are Pτ G({red, blue}) = X and P G({red, blue}) = X.

Thus, the Tolerance Superhyperrough Set is the collection of pairs:

A = ∅ : (∅, ∅), A = {red} : (∅, {1, 2, 3, 4}), A = {blue} : (∅, {1, 2, 3, 4}), A = {red, blue} : (X, X).

We define Dynamic Hyperrough Set as follows.

[Dynamic Hyperrough Set] Let T1, T2, . . . , Tn be n distinct attributes with corresponding domains J1, J2, . . . , Jn. Define the Cartesian product J = J1 × J2 × · · · × Jn.

Assume there is a mapping F : J → P(X) that assigns to each attribute combination a = (a1, . . . , an) ∈ J a subset F(a) ⊆ X. In order to capture dynamic effects, assume that for every a ∈ J there exist two transfer functions ρa : X → [0, 1] and ρa : X → [0, 1], together with fixed thresholds da , da ∈ [0, 1]. Then for each a ∈ J define the inward (addition) set and the outward (removal) set by I(a) = {x ∈ X \ F(a) : ρa (x) ≥ da }, C(a) = {x ∈ F(a) : ρa (x) ≥ da }.

The dynamic modified set associated with a is D(a) = F(a) ∪ I(a) \ C(a).

The pair (D, J), where D : J → P(X) is given by a 7→ D(a), is called a dynamic Hyperrough Set. Its dynamic approximations are given by D(a) = {x ∈ X : [x]R ⊆ D(a)} and D(a) = {x ∈ X : [x]R ∩ D(a) ̸= ∅}.

[Dynamic Hyperrough Set] Let X = {u1, u2, u3, u4, u5} be a finite universe. Consider two attributes:

T1 : Color with domain J1 = {Red, Blue}, T2 : Shape with domain J2 = {Circle, Square}.

Then the attribute combination domain is J = J1 × J2.

Define the mapping F : J → P(X) by setting, for instance, F(Red, Circle) = {u1, u2}, F(Red, Square) = {u3}, F(Blue, Circle) = {u2, u4}, F(Blue, Square) = {u5}.

To incorporate dynamic effects, for each attribute combination a ∈ J we introduce two transfer functions: ρa : X → [0, 1] (inward transfer), ρa : X → [0, 1] (outward transfer).

For the combination a = (Red, Circle), suppose that the dynamic transfer values are assigned as follows: 0.8, if x ∈ {u3, u4}, ρ (x) =

(Red,Circle)

0.5, otherwise, 0.6, if x ∈ {u2}, ρ (x) =

(Red,Circle)

0.3, otherwise.

Let the thresholds be d = 0.7 and d = 0.5.

(Red,Circle) (Red,Circle)

Then we define the inward (addition) set and the outward (removal) set for a = (Red, Circle) by I(a) = {x ∈ X \ F(a) : ρa (x) ≥ da } = {u3, u4}, C(a) = {x ∈ F(a) : ρa (x) ≥ da } = {u2}.

Hence, the dynamic modified set is computed as D(a) = F(a) ∪ I(a) \ C(a) = {u1, u2} ∪ {u3, u4} \ {u2} = {u1, u3, u4}.

Now, assume that an equivalence relation R ⊆ X × X is given by partitioning X into [u1]R = {u1, u3, u4} and [u2]R = {u2, u5}.

Then the dynamic lower and upper approximations for the combination (Red, Circle) are D(a) = {x ∈ X : [x]R ⊆ D(a)} = {u1, u3, u4},

D(a) = {x ∈ X : [x]R ∩ D(a) ̸= ∅} = X.

Thus, this example concretely illustrates the computation of a Dynamic Hyperrough Set.

[Dynamic Hyperrough Set Generalizes Hyperrough Set and Dynamic Rough Set] The dynamic Hyperrough Set (D, J) defined in Definition generalizes:

(1) The classical Hyperrough Set: If for every a ∈ J and for every x ∈ X the dynamic conditions are vacuous, i.e.

+ + − −

ρa (x) < da and ρa (x) < da , then I(a) = ∅ and C(a) = ∅, so that D(a) = F(a).

(2) The dynamic Rough Set: In the special case when n = 1 (so that J = J1), the definition reduces to that of the dynamic Rough Set.

Proof : (1) Recovery of the Classical Hyperrough Set: Assume that for all a ∈ J and every x ∈ X, the inequalities

+ + − −

ρa (x) < da and ρa (x) < da hold. Then by definition, I(a) = {x ∈ X \ F(a) : ρa (x) ≥ da } = ∅, and C(a) = {x ∈ F(a) : ρa (x) ≥ da } = ∅.

D(a) = (F(a) ∪ ∅) \ ∅ = F(a), for every a ∈ J. Hence, the dynamic construction yields exactly the classical Hyperrough Set.

(2) Recovery of the Dynamic Rough Set: When n = 1, the Cartesian product reduces to J = J1 and the mapping F : J1 → P(X) assigns subsets of X based on a single attribute value. In this case, the dynamic procedure of forming D(a) = F(a) ∪ I(a) \ C(a) is exactly that adopted in the dynamic Rough Set framework. Therefore, the dynamic Hyperrough Set generalizes the dynamic Rough Set when there is only one attribute. □

We define Dynamic SuperHyperrough Set as follows.

[Dynamic SuperHyperrough Set] Let T1, T2, . . . , Tn be n distinct attributes with corresponding domains J1, J2, . . . , Jn. For each attribute Ti, let P(Ji) denote its power set. Define J = P(J1) × P(J2) × · · · × P(Jn).

Assume there is a mapping F : J → P(X) assigning to each combination A = (A1, A2, . . . , An) with Ai ⊆ Ji, a subset F(A) ⊆ X. In order to incorporate dynamics, suppose that for every A ∈ J there exist dynamic transfer functions ρ : X → [0, 1] and ρ : X → [0, 1], with thresholds d , d ∈ [0, 1]. Then define

+ + − −

I(A) = {x ∈ X \ F(A) : ρ (x) ≥ d }, C(A) = {x ∈ F(A) : ρ (x) ≥ d }.

A A A A

The dynamic modified set for A is given by D(A) = F(A) ∪ I(A) \ C(A).

Thus, the pair (D, J) with D : J → P(X) defined by A 7→ D(A) is called an Dynamic SuperHyperrough Set. Its lower and upper approximations are defined as D(A) = {x ∈ X : [x]R ⊆ D(A)}, D(A) = {x ∈ X : [x]R ∩ D(A) ̸= ∅}.

[Dynamic SuperHyperrough Set] Let X = {u1, u2, u3, u4} be a finite universe. Consider two attributes:

T1 : Type with domain J1 = {A, B}, T2 : Category with domain J2 = {X, Y }.

For each attribute, we consider its power set. Hence, define P(J1) = {∅, {A}, {B}, {A, B}}, P(J2) = {∅, {X}, {Y }, {X, Y }}.

Then, the attribute combination domain is given by J = P(J1) × P(J2).

Define a mapping F : J → P(X) as follows:

F({A}, {X}) = {u1, u2}, F({B}, {X}) = {u3}, F({A}, {Y }) = {u2, u4}, F({A, B}, {X, Y }) = X, and for all other combinations, let F(A) = ∅.

To introduce dynamic modifications, assign for each combination A ∈ J appropriate transfer functions. For instance, for the combination A = ({A}, {X}), assume that ( 0.9, if x ∈ {u3, u4}, ρ (x) = 0.4, otherwise, and ( 0.7, if x ∈ {u2}, ρ (x) = 0.3, otherwise.

Choose thresholds d = 0.8 and d = 0.6.

Then, for A = ({A}, {X}), we compute:

I(A) = {x ∈ X \ F(A) : ρ (x) ≥ d } = {u3, u4}, C(A) = {x ∈ F(A) : ρ (x) ≥ d } = {u2}.

Thus, the dynamic modified set for A is D(A) = F(A) ∪ I(A) \ C(A) = {u1, u2} ∪ {u3, u4} \ {u2} = {u1, u3, u4}.

Suppose now that the equivalence relation R on X partitions the universe as [u1]R = {u1, u3} and [u2]R = {u2, u4}.

Then the dynamic approximations for A = ({A}, {X}) are given by D(A) = {x ∈ X : [x]R ⊆ D(A)} = {u1, u3}, D(A) = {x ∈ X : [x]R ∩ D(A) ̸= ∅} = X.

This example concretely demonstrates the computation within a Dynamic SuperHyperrough Set, where the domain of attribute combinations is expanded to include subsets of the original attribute domains.

[Dynamic SuperHyperrough Set Generalizes SuperHyperrough Set and Dynamic Hyperrough Set] The dynamic SuperHyperrough Set (D, J) defined in Definition generalizes:

(1) The classical SuperHyperrough Set: If for every A ∈ J and every x ∈ X,

+ + − −

ρ (x) < d and ρ (x) < d ,

A A A A

then I(A) = ∅ and C(A) = ∅, so that D(A) = F(A).

(2) The dynamic Hyperrough Set: If we restrict J to the subset J = {({a1}, {a2}, . . . , {an}) : ai ∈ Ji}, then the mapping F on J and the corresponding dynamic modification coincide with the ones in Definition ; hence, the dynamic SuperHyperrough Set reduces to the dynamic Hyperrough Set.

Proof : (1) Recovery of the Classical SuperHyperrough Set: Assume that for all A ∈ J and every x ∈ X the conditions

+ + − −

ρ (x) < d and ρ (x) < d

A A A A

are satisfied. Then, by definition, I(A) = ∅ and C(A) = ∅.

D(A) = F(A) ∪ ∅ \ ∅ = F(A).

It follows that the dynamic modification disappears, and the structure coincides with that of a classical SuperHyperrough Set.

(2) Reduction to the Dynamic Hyperrough Set: Now, consider the subset J = {({a1}, {a2}, . . . , {an}) : ai ∈ Ji} ⊆ J.

Define the mapping F on J by F ({a1}, {a2}, . . . , {an}) := F(a1, a2, . . . , an).

For each element of J , the dynamic transfer functions and thresholds (restricted to singleton sets) are taken to be identical to those used in the dynamic Hyperrough Set (Definition ). Hence, for every A = ({a1}, . . . , {an}) ∈ J one obtains D(A) = F (A) ∪ I(A) \ C(A) which is exactly the same as in the dynamic Hyperrough Set definition. Therefore, under the restriction to singleton components, the dynamic SuperHyperrough Set reduces to the dynamic Hyperrough Set. □

The authors hereby confirm that, to the best of their knowledge, this manuscript is their original work, has not been published in any other journal, and is not currently under consideration for publication elsewhere at this stage.