On rough sets induced by fuzzy relations approach in semigroups

Definition 2.1

[19] A fuzzy set of U is defined as a function from U to the closed unit interval [0, 1].

Definition 2.2

[22] Let 𝓕(U × V) be a family of all fuzzy sets of U × V. An element in 𝓕(U × V) is referred to as a fuzzy relation fromUtoV. An element in 𝓕(U × V) is called a fuzzy relation onU if U = V. For a fuzzy relation Θ ∈ 𝓕(U × V) and elements u ∈ U, v ∈ V, the value of Θ(u, v) in [0, 1] representing the membership grade of relation betweenuandvunderΘ. If Θ ∈ 𝓕(U × V) where U := {u₁, u₂, u₃, …, u_m} and V := {v₁, v₂, v₃, …, v_n}, then the fuzzy relation Θ is represented by the matrix as

Definition 2.3

[22] Let Θ be a fuzzy relation from U to V. Θ is called serial if for all u ∈ U, there exists v ∈ V such that Θ(u, v) = 1.

Definition 2.4

[22] Let Θ be a fuzzy relation on U.

1. Θ is called reflexive if for all u ∈ U, Θ(u, u) = 1,

2. Θ is called symmetric if for all u₁, u₂ ∈ U, Θ(u₁, u₂) = Θ(u₂, u₁),

3. Θ is called transitive if for all u₁, u₂ ∈ U, Θ(u₁, u₂) ≥ ∨_u3∈U (Θ(u₁, u₃) ∧ Θ(u₃, u₂)),

4. Θ is called a similarity fuzzy relation if it is reflexive, symmetric and transitive.

Definition 2.5

[25] Let Θ be a fuzzy relations on S. Θ is called compatible if for all s₁, s₂, s₃ ∈ S,

Definition 3.1

Let ι ∈ [0, 1] and let Θ be a fuzzy relation from U to V. For an element u ∈ U,

is called a successor class ofuwith respect toι-level underΘ.

Remark 3.2

Let ι ∈ [0, 1]. If Θ is a serial fuzzy relation from U to V, then S_Θ(u; ι) ≠ ∅ for all u ∈ U.

Definition 3.3

Let ι ∈ [0, 1] and let Θ be a fuzzy relation from U to V. For an element u₁ ∈ U,

is called a core of the successor class ofu₁with respect toι-level underΘ.

We denote by 𝓒𝓢_Θ(U; ι) the collection of CS_Θ(u; ι) for all u ∈ U.

Proposition 3.4

Letι ∈ [0, 1] and letΘbe a fuzzy relation fromUtoV. Then the following statements hold.

1. For allu ∈ U, u ∈ CS_Θ(u; ι).

2. For allu₁, u₂ ∈ U, u₂ ∈ CS_Θ(u₁; ι) if and only ifCS_Θ(u₁; ι) = CS_Θ(u₂; ι).

Remark 3.5

Let ι ∈ [0, 1] and let Θ be a fuzzy relation from U to V. Then 𝓒𝓢_Θ(U; ι) is the partition of U.

Proposition 3.6

Letι ∈ [0, 1] and letΘbe a fuzzy relation onU. Then we have the following statements.

1. IfΘis reflexive, thenCS_Θ(u; ι) ⊆ S_Θ(u; ι) for allu ∈ U.

2. IfΘis a similarity fuzzy relation, thenS_Θ(u; ι) andCS_Θ(u; ι) are identical classes for allu ∈ U.

Proof

The proof is straightforward, so we omit it. □

Definition 3.7

Let ι ∈ [0, 1] and let Θ be a fuzzy relation from U to V. A triple (U, V, 𝓒𝓢_Θ(U; ι)) is called an approximation space based on 𝓒𝓢_Θ(U; ι) (briefly, 𝓒𝓢_Θ(U; ι)-approximation space). If U = V, then (U, V, 𝓒𝓢_Θ(U; ι)) is replaced by a pair (U, 𝓒𝓢_Θ(U; ι)).

Definition 3.8

Let (U, V, 𝓒𝓢_Θ(U; ι)) be an 𝓒𝓢_Θ(U; ι)-approximation space. For a non-empty subset X of U, we define three sets as follows:

Θ(X; ι) := ⋃_u∈U{CS_Θ(u; ι) : CS_Θ(u; ι) ∩ X ≠ ∅},

Θ(X; ι) := ⋃_u∈U{CS_Θ(u; ι) : CS_Θ(u; ι) ⊆ X} and

Θ_bnd(X; ι) := Θ(X; ι) − Θ(X; ι).

Then

1. Θ(X; ι) is called an upper approximation ofXin (U, V, 𝓒𝓢_Θ(U; ι))

   (briefly, 𝓒𝓢_Θ(U; ι)-upper approximation ofX).

2. Θ(X; ι) is called a lower approximation ofXin (U, V, 𝓒𝓢_Θ(U; ι))

   (briefly, 𝓒𝓢_Θ(U; ι)-lower approximation ofX).

3. Θ_bnd(X; ι) is called a boundary region ofXin (U, V, 𝓒𝓢_Θ(U; ι))

   (briefly, 𝓒𝓢_Θ(U; ι)-boundary region ofX).

4. If Θ_bnd(X; ι) ≠ ∅, then ΘR̦ (X; ι) := (Θ(X; ι), Θ(X; ι)) is called a rough set ofXin (U, V, 𝓒𝓢_Θ(U; ι))

   (briefly, 𝓒𝓢_Θ(U; ι)-rough set ofX).

5. If Θ_bnd(X; ι) = ∅, then X is called a definable set in (U, V, 𝓒𝓢_Θ(U; ι))

   (briefly, 𝓒𝓢_Θ(U; ι)-definable set).

Example 3.9

Let U = {u₁, u₂, u₃, u₄, u₅} be a set of doctoral students in a mathematical business classroom of a university and let V = {v₁, v₂, v₃, v₄} be a set of subjects where

v₁ is business,

v₂ is economics,

v₃ is computer sciences and

v₄ is mathematics.

For a fuzzy relation Θ ∈ 𝓕(U × V) and elements u ∈ U, v ∈ V, the number Θ(u, v) in the closed unit interval [0, 1] is defined as the score of the doctoral student u with respect to the subject v under Θ. The scores of all doctoral students in U with respect to subjects in V under Θ are given as the following matrix.

Let ι = 0.9 be a minimal score level. If an educational measurement committee assign X := {u₂, u₃, u₅} which is a set of excellent doctoral students under the global evaluation, then the assessment of X in an 𝓒𝓢_Θ(U; 0.9)-approximation space (U, V, 𝓒𝓢_Θ(U; 0.9)) is derived by the process as the following.

According to Definition 3.1, it follows that

S_Θ(u₁; 0.9) := {v₂, v₄},

S_Θ(u₂; 0.9) := {v₂, v₄},

S_Θ(u₃; 0.9) := {v₁, v₄},

S_Θ(u₄; 0.9) := {v₃, v₄} and

S_Θ(u₅; 0.9) := {v₁, v₂, v₄}.

According to Definition 3.3, it follows that

CS_Θ(u₁; 0.9) := {u₁, u₂},

CS_Θ(u₂; 0.9) := {u₁, u₂},

CS_Θ(u₃; 0.9) := {u₃},

CS_Θ(u₄; 0.9) := {u₄} and

CS_Θ(u₅; 0.9) := {u₅}.

According to Definition 3.8, it follows that

Θ(X; 0.9) := {u₁, u₂, u₃, u₅},

Θ(X; 0.9) := {u₃, u₅} and

Θ_bnd(X; 0.9) := {u₁, u₂}.

Therefore Θ R̦(X; 0.9) := ({u₁, u₂, u₃, u₅}, {u₃, u₅}) is a 𝓒𝓢_Θ(U; 0.9)-rough set of X. Consequently,

1. u₁, u₂, u₃ and u₅ are possibly excellent doctoral students,

2. u₃ and u₅ are certainly excellent doctoral students and

3. for u₁ and u₂ it cannot be determined whether two students are excellent doctoral students or not.

Definition 3.10

Let (U, V, 𝓒𝓢_Θ(U; ι)) be an 𝓒𝓢_Θ(U; ι)-approximation space and let X be a non-empty subset of U. Θ(X; ι) is called a non-empty 𝓒𝓢_Θ(U; ι)-upper approximation ofXin (U, V, 𝓒𝓢_Θ(U; ι)) if Θ(X; ι) is a non-empty subset of U. Similarly, we can define a non-empty 𝓒𝓢_Θ(U; ι)-lower approximation. Θ R̦(X; ι) is referred to as a non-empty 𝓒𝓢_Θ(U; ι)-rough set in (U, V, 𝓒𝓢_Θ(U; ι)) if Θ(X; ι) is a non-empty 𝓒𝓢_Θ(U; ι)-upper approximation and Θ(X; ι) is a non-empty 𝓒𝓢_Θ(U; ι)-lower approximation.

Proposition 3.11

Let (U, V, 𝓒𝓢_Θ(U; ι)) be an 𝓒𝓢_Θ(U; ι)-approximation space. IfXandYare non-empty subsets ofU, then we have the following statements.

1. Θ(U; ι) = Uand

   Θ(U; ι) = U.

2. Θ(∅; ι) = ∅ and

   Θ(∅; ι) = ∅.

3. X ⊆ Θ(X; ι) and

   Θ(X; ι) ⊆ X.

4. Θ(X ∪ Y; ι) = Θ(X; ι) ∪ Θ(Y; ι) and

   Θ(X ∩ Y; ι) = Θ(X; ι) ∩ Θ(Y; ι).

5. Θ(X ∩ Y; ι) ⊆ Θ(X; ι) ∩ Θ(Y; ι) and

   Θ(X ∪ Y; ι) ⊇ Θ(X; ι) ∪ Θ(Y; ι).

6. Θ(X^c; ι) = (Θ(X; ι))^c, whereX^cand (Θ(X; ι))^care complements ofXandΘ(X; ι), respectively.

7. Θ(Θ(X; ι); ι) = Θ(X; ι) and

   Θ(Θ(X; ι); ι) = Θ(X; ι).

8. Θ((Θ(X; ι))^c; ι) = (Θ(X; ι))^c, where (Θ(X; ι))^cis a complement ofΘ(X; ι) and

   Θ((Θ(X; ι))^c; ι) = (Θ(X; ι))^c, where (Θ(X; ι))^cis a complement ofΘ(X; ι).

9. If X ⊆ Y, thenΘ(X; ι) ⊆ Θ(Y; ι) andΘ(X; ι) ⊆ Θ(Y; ι).

Proof

The proof is straightforward, so we omit it. □

Definition 3.12

Let (U, V, 𝓒𝓢_Θ(U; ι)) be an 𝓒𝓢_Θ(U; ι)-approximation space and let X be a non-empty subset of U. If Θ(X; ι) is a non-empty 𝓒𝓢_Θ(U; ι)-lower approximation of X in (U, V, 𝓒𝓢_Θ(U; ι)) and Θ(X; ι) is a proper subset of X, then X is called a set over a non-empty interior set.

Proposition 3.13

Let (U, V, 𝓒𝓢_Θ(U; ι)) be an 𝓒𝓢_Θ(U; ι)-approximation space and letXbe a non-empty subset ofU. IfXis a set over non-empty interior set, thenΘ R̦(X; ι) is a non-empty 𝓒𝓢_Θ(U; ι)-rough set ofXin (U, V, 𝓒𝓢_Θ(U; ι)).

Proof

Suppose that X is a set over a non-empty interior set. Then we have that Θ(X; ι) is a non-empty 𝓒𝓢_Θ(U; ι)-lower approximation and Θ(X; ι) ⊂ X. By Proposition 3.11 (3), we obtain that ∅ ≠ X ⊆ Θ(X; ι). Thus we get Θ(X; ι) is a non-empty 𝓒𝓢_Θ(U; ι)-upper approximation. We shall verify that Θ_bnd(X; ι) ≠ ∅. Suppose that Θ_bnd(X; ι) = ∅. Then we have Θ(X; ι) = Θ(X; ι). From Proposition 3.11 (3), once again, it follows that Θ(X; ι) = X, a contradiction. Therefore Θ_bnd(X; ι) ≠ ∅. Consequently, Θ(X; ι) is a non-empty 𝓒𝓢_Θ(U; ι)-rough set of X. □

Example 3.14

Let U := {u₁ = 3, u₂ = 1, u3=13,u4=19,u5=127 } and V := {v₁ = 2, v₂ = 2 3 , v₃ = 6, v₄ = 6 3 }. Define a fuzzy relation Θ ∈ 𝓕(U × V) by

for all (u, v) ∈ U × V. Then we have the following ranges of Θ.

Let ι = 0.95 and let X := {u₂, u₃} be a non-empty subset of U. According to Definition 3.1, it follows that

S_Θ(u₁; 0.95) := {v₁},

S_Θ(u₂; 0.95) := {v₁},

S_Θ(u₃; 0.95) := {v₁, v₂, v₃},

S_Θ(u₄; 0.95) := {v₁, v₂, v₃, v₄} and

S_Θ(u₅; 0.95) := {v₁, v₂, v₃, v₄}.

According to Definition 3.3, it follows that

CS_Θ(u₁; 0.95) := {u₁, u₂},

CS_Θ(u₂; 0.95) := {u₁, u₂},

CS_Θ(u₃; 0.95) := {u₃},

CS_Θ(u₄; 0.95) := {u₄, u₅} and

CS_Θ(u₅; 0.95) := {u₄, u₅}.

Here it is easy to check that Θ(X; 0.95) is a non-empty 𝓒𝓢_Θ(U; 95)-lower approximation of X, and also Θ(X; 0.95) ⊂ X. Note that X ⊆ Θ(X; 0.95). Thus we get Θ(X; 0.95) ≠ ∅ and Θ(X; 0.95) ≠ Θ(X; 0.95). It follows that Θ R̦(X; 0.95) is a non-empty 𝓒𝓢_Θ(U; 0.95)-rough set of X.

Proposition 3.15

Let (U, 𝓒𝓢_Θ(U; ι)) be an 𝓒𝓢_Θ(U; ι)-approximation space and let (U, 𝓒𝓢_Ψ(U; κ)) be an 𝓒𝓢_Ψ(U; κ)-approximation space. Ifι ≥ κandΘ ⊆ ΨwhereΘis reflexive andΨis transitive, then we haveΘ(X; ι) ⊆ Ψ(X; κ) for every non-empty subsetXofU.

Proof

Let X be a non-empty subset of U. Then we prove that Θ(X; ι) ⊆ Ψ(X; κ). In fact, let u₁ ∈ Θ(X; ι). Then CS_Θ(u₁; ι) ∩ X ≠ ∅. Thus there exists u₂ ∈ CS_Θ(u₁; ι) ∩ X, and so S_Θ(u₁; ι) = S_Θ(u₂; ι). Since Θ is reflexive, we have Θ(u₂, u₂) = 1 ≥ ι. Whence u₂ ∈ S_Θ(u₂; ι) = S_Θ(u₁; ι). Thus we have Θ(u₁, u₂) ≥ ι. Since ι ≥ κ and Θ ⊆ Ψ, we have Ψ(u₁, u₂) ≥ Θ(u₁, u₂) ≥ κ, and so Ψ(u₁, u₂) ≥ κ. Similary, we have Ψ(u₂, u₁) ≥ κ. We shall verify that S_Ψ(u₁; κ) = S_Ψ(u₂; κ). Now, let u₃ ∈ S_Ψ(u₂;κ). Then Ψ(u₂, u₃) ≥ κ. Since Ψ is transitive, we have

Hence Ψ(u₁, u₃) ≥ κ. Thus u₃ ∈ S_Ψ(u₁; κ), which yields S_Ψ(u₂; κ) ⊆ S_Ψ(u₁; κ). Similary, we can prove that S_Ψ(u₁; κ) ⊆ S_Ψ(u₂; κ). Whence we get S_Ψ(u₁; κ) = S_Ψ(u₂; κ), and so u₂ ∈ CS_Ψ(u₁; κ). Thus we have that u₂ ∈ CS_Ψ(u₁; κ) ∩ X. Hence CS_Ψ(u₁; κ) ∩ X ≠ ∅, which yields u₁ ∈ Ψ(X; κ). Therefore we get that Θ(X; ι) ⊆ Ψ(X; κ). □

Proposition 3.16

Let (U, 𝓒𝓢_Θ(U; ι)) be an 𝓒𝓢_Θ(U; ι)-approximation space and let (U, 𝓒𝓢_Ψ(U; κ)) be an 𝓒𝓢_Ψ(U; κ)-approximation space. Ifι ≥ κandΘ ⊆ ΨwhereΘis reflexive andΨis transitive, then we haveΨ(X; κ) ⊆ Θ(X; ι) for every non-empty subsetX of U.

Proof

Let X be a non-empty subset of U. Then we prove that Ψ(X; κ) ⊆ Θ(X; ι). Indeed, let u₁ ∈ Ψ(X; κ). Then CS_Ψ(u₁; ι) ⊆ X. We shall show that CS_Θ(u₁; ι) ⊆ CS_Ψ(u₁; κ). Let u₂ ∈ CS_Θ(u₁; ι). Then we have S_Θ(u₁; ι) = S_Θ(u₂; ι). Since Θ is reflexive, we have that Θ(u₁, u₁) = 1 ≥ ι. Hence u₁ ∈ S_Θ(u₁; ι), and so u₁ ∈ S_Θ(u₂; ι). Thus Θ(u₂, u₁) ≥ ι. By the assumption, we have Ψ(u₂, u₁) ≥ Θ(u₂, u₁) ≥ κ, and so Ψ(u₂, u₁) ≥ κ. Similary, we get that Ψ(u₁, u₂) ≥ κ. We shall prove that S_Ψ(u₁; κ) = S_Ψ(u₂; κ). Let u₃ ∈ S_Ψ(u₂; κ). Then Ψ(u₂, u₃) ≥ κ. Since Ψ is transitive, we have

Thus Ψ(u₁, u₃) ≥ κ, and so u₃ ∈ S_Ψ(u₁; κ). Hence S_Ψ(u₂; κ) ⊆ S_Ψ(u₁; κ). Similary, we can prove that S_Ψ(u₁; κ) ⊆ S_Ψ(u₂; κ), which yields S_Ψ(u₁; κ) = S_Ψ(u₂; κ). Thus we have u₂ ∈ CS_Ψ(u₁; κ), and so CS_Θ(u₁; ι) ⊆ CS_Ψ(u₁; κ) ⊆ X. Therefore u₁ ∈ Θ(X; ι). This means that Ψ(X; κ) ⊆ Θ(X; ι). □

Definition 4.1

Let Θ be a fuzzy relation on S. Θ is called a compatible preorder fuzzy relation if Θ is reflexive, transitive and compatible. An 𝓒𝓢_Θ(S; ι)-approximation space (S, 𝓒𝓢_Θ(S; ι)) is called an 𝓒𝓢_Θ(S; ι)-approximation space type CPF if Θ is a compatible preorder fuzzy relation.

Proposition 4.2

If (S, 𝓒𝓢_Θ(S; ι)) is an 𝓒𝓢_Θ(S; ι)-approximation space type CPF, then

for alls₁, s₂ ∈ S.

Proof

Let s₁, s₂ be two elements in S and let s₃ ∈ (CS_Θ(s₁; ι)) (CS_Θ(s₂; ι)). Then there exist s₄ ∈ CS_Θ(s₁; ι) and s₅ ∈ CS_Θ(s₂; ι) such that s₃ = s₄s₅. Thus S_Θ(s₁; ι) = S_Θ(s₄; ι) and S_Θ(s₂; ι) = S_Θ(s₅; ι). Hence we get that S_Θ(s₁s₂; ι) = S_Θ(s₄s₅; ι). Indeed, we suppose that s₆ ∈ S_Θ(s₄s₅; ι). Then we have Θ(s₄s₅, s₆) ≥ ι. Since Θ is reflexive, we have Θ(s₄, s₄) = Θ(s₅, s₅) = 1 ≥ ι, and so s₄ ∈ S_Θ(s₄; ι) and s₅ ∈ S_Θ(s₅; ι). Whence s₄ ∈ S_Θ(s₁; ι) and s₅ ∈ S_Θ(s₂; ι). Thus Θ(s₁, s₄) ≥ ι and Θ(s₂, s₅) ≥ ι. Since Θ is transitive and compatible, we have

Hence Θ(s₁s₂, s₄s₅) ≥ ι. Since Θ is transitive, we have

Thus Θ(s₁s₂, s₆) ≥ ι, and so s₆ ∈ S_Θ(s₁s₂; ι). Hence S_Θ(s₄s₅; ι) ⊆ S_Θ(s₁s₂; ι). Similarly, we can show that S_Θ(s₁s₂; ι) ⊆ S_Θ(s₄s₅; ι). Thus S_Θ(s₁s₂; ι) = S_Θ(s₄s₅; ι), which yields s₃ ∈ CS_Θ(s₁s₂; ι). This implies that (CS_Θ(s₁; ι)) (CS_Θ(s₂; ι)) ⊆ CS_Θ(s₁s₂; ι). □

Example 4.3

Let S := {s₁, s₂, s₃, s₄, s₅} be a semigroup with multiplication rules defined by Table 1.

Define the membership grades of relationship between any two elements in S under the fuzzy relation Θ on S as the following.

Then it is easy to check that Θ is a compatible preorder fuzzy relation. For ι = 0.9, successor classes of each elements in S with respect to 0.9-level under Θ are

S_Θ(s₁; 0.9) := {s₁, s₃, s₅},

S_Θ(s₂; 0.9) := {s₂},

S_Θ(s₃; 0.9) := {s₃, s₅},

S_Θ(s₄; 0.9) := {s₄} and

S_Θ(s₅; 0.9) := {s₃, s₅}.

Hence cores of successor classes of each elements in S with respect to 0.9-level under Θ are

CS_Θ(s₁; 0.9) := {s₁},

CS_Θ(s₂; 0.9) := {s₂},

CS_Θ(s₃; 0.9) := {s₃, s₅},

CS_Θ(s₄; 0.9) := {s₄} and

CS_Θ(s₅; 0.9) := {s₃, s₅}.

Here it is straightforward to verify that (CS_Θ(s; 0.9))(CS_Θ(s′; 0.9)) ⊆ CS_Θ(ss′; 0.9) for all s, s′ ∈ S.

Example 4.4

Let S := {s₁, s₂, s₃, s₄, s₅} be a semigroup with multiplication rules defined by Table 2.

Define the membership grades of relationship between any two elements in S under the fuzzy relation Θ on S as the following.

Then it is easy to check that Θ is a compatible preorder fuzzy relation. For ι = 0.9, successor classes of each elements in S with respect to 0.9-level under Θ are

S_Θ(s₁; 0.9) := {s₁, s₅},

S_Θ(s₂; 0.9) := {s₂, s₃, s₄},

S_Θ(s₃; 0.9) := {s₂, s₃, s₄},

S_Θ(s₄; 0.9) := {s₂, s₃, s₄} and

S_Θ(s₅; 0.9) := {s₅}.

Hence cores of successor classes of each elements in S with respect to 0.9-level under Θ are

CS_Θ(s₁; 0.9) := {s₁},

CS_Θ(s₂; 0.9) := {s₂, s₃, s₄},

CS_Θ(s₃; 0.9) := {s₂, s₃, s₄},

CS_Θ(s₄; 0.9) := {s₂, s₃, s₄} and

CS_Θ(s₅; 0.9) := {s₅}.

Here it is straightforward to check that (CS_Θ(s; 0.9))(CS_Θ(s′; 0.9)) = CS_Θ(ss′; 0.9) for all s, s′ ∈ S. Based on this point, the property can be considered as a special case of Proposition 4.2. This example leads to the following definition.

Definition 4.5

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CPF. The collection 𝓒𝓢_Θ(S; ι) is called complete induced byΘ (briefly, Θ-complete) if for all s₁, s₂ ∈ S,

Definition 4.6

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CPF. If 𝓒𝓢_Θ(S; ι) is complete induced by Θ, then Θ is called a complete fuzzy relation. (S, 𝓒𝓢_Θ(S; ι)) is called an 𝓒𝓢_Θ(S; ι)-approximation space type CF if Θ is complete.

Proposition 4.7

If (S, 𝓒𝓢_Θ(S; ι)) is an 𝓒𝓢_Θ(S; ι)-approximation space type CPF, then

for every non-empty subsets X, YofS.

Proof

Let X and Y be two non-empty subsets of S. Suppose that s₁ ∈ (Θ(X; ι))(Θ(Y; ι)). Then there exist s₂ ∈ Θ(X; ι) and s₃ ∈ Θ(Y; ι) such that s₁ = s₂s₃. Thus we have that CS_Θ(s₂; ι) ∩ X ≠ ∅ and CS_Θ(s₃; ι) ∩ Y ≠ ∅. Then there exist s₄, s₅ ∈ S such that s₄ ∈ CS_Θ(s₂; ι) ∩ X and s₅ ∈ CS_Θ(s₃; ι) ∩ Y. From Proposition 4.2, it follows that s₄s₅ ∈ (CS_Θ(s₂; ι))(CS_Θ(s₃; ι)) ⊆ CS_Θ(s₂s₃; ι) and s₄s₅ ∈ XY. Thus CS_Θ(s₂s₃; ι) ∩ XY ≠ ∅, which yields s₁ = s₂s₃ ∈ Θ(XY; ι). Therefore (Θ(X; ι))( Θ(Y; ι)) ⊆ Θ(XY; ι). □

Proposition 4.8

If (S, 𝓒𝓢_Θ(S; ι)) is an 𝓒𝓢_Θ(S; ι)-approximation space type CF, then

for every non-empty subsets X, YofS.

Proof

Let X and Y be two non-empty subsets of S and let s₁ ∈ (Θ(X; ι))(Θ(Y; ι)). Then there exist s₂ ∈ Θ(X; ι) and s₃ ∈ Θ(Y; ι) such that s₁ = s₂s₃, and so CS_Θ(s₂; ι) ⊆ X and CS_Θ(s₃; ι) ⊆ Y. Since Θ is complete, we get CS_Θ(s₂s₃; ι) = CS_Θ(s₂; ι)CS_Θ(s₃; ι) ⊆ XY. Thus CS_Θ(s₂s₃; ι) ⊆ XY. Hence s₁ = s₂s₃ ∈ Θ(XY; ι). Therefore (Θ(X; ι))(Θ(Y; ι)) ⊆ Θ(XY; ι). □

Example 4.9

According to Example 4.4, suppose that X := {s₁, s₄, s₅} is a subset of S. Then we have Θ(X; ι) = S and Θ(X; ι) := {s₁, s₅}. Here it is easy to verify that Θ(X; ι) and Θ(X; ι) are subsemigroups, ideals and completely prime ideals of S. Moreover, we also have Θ_bnd(X; ι) is a non-empty set. For the existence of subsemigroups, ideals and completely prime ideals of S under compatible preorder fuzzy relations in this example, we give the following definition.

Definition 4.10

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CPF and let X be a non-empty subset of S. A non-empty 𝓒𝓢_Θ(S; ι)-upper approximation Θ(X; ι) of X in (S, 𝓒𝓢_Θ(S; ι)) is called an 𝓒𝓢_Θ(S; ι)-upper approximation semigroup if it is a subsemigroup of S. A non-empty 𝓒𝓢_Θ(S; ι)-lower approximation Θ(X; ι) of X in (S, 𝓒𝓢_Θ(S; ι)) is called a 𝓒𝓢_Θ(S; ι)-lower approximation semigroup if it is a subsemigroup of S. A non-empty 𝓒𝓢_Θ(S; ι)-rough set Θ R̦(X; ι) of X in (S, 𝓒𝓢_Θ(S; ι)) is called a 𝓒𝓢_Θ(S; ι)-rough semigroup if Θ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation semigroup and Θ(X; ι) is a 𝓒𝓢_Θ(S; ι)-lower approximation semigroup.

Similarly, we can define 𝓒𝓢_Θ(S; ι)-rough (completely prime) ideals.

Theorem 4.11

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CPF. IfXis a subsemigroup ofS, thenΘ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation semigroup.

Proof

Suppose that X is a subsemigroup of S. Then XX ⊆ X. By Proposition 3.11 (3), we obtain that ∅ ≠ X ⊆ Θ(X; ι). Hence Θ(X; ι) is a non-empty 𝓒𝓢_Θ(S; ι)-upper approximation. From Proposition 3.11 (9), it follows that Θ(XX; ι) ⊆ Θ(X; ι). By Proposition 4.7, we obtain that

Hence Θ(X; ι) is a subsemigroup of S. Thus Θ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation semigroup. □

Theorem 4.12

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CF. IfXis a subsemigroup ofSwithΘ(X; ι) ≠ ∅, thenΘ(X; ι) is a 𝓒𝓢_Θ(S; ι)-lower approximation semigroup.

Proof

Suppose that X is a subsemigroup of S. Then XX ⊆ X. Obviously, Θ(X; ι) is a non-empty 𝓒𝓢_Θ(S; ι)-lower approximation. From Proposition 3.11 (9), it follows that Θ(XX; ι) ⊆ Θ(X; ι). By Proposition 4.8, we obtain that

Thus Θ(X; ι) is a subsemigroup of S. Therefore Θ(X; ι) is a 𝓒𝓢_Θ(S; ι)-lower approximation semigroup. □

Corollary 4.13

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CF. IfXis a subsemigroup ofSover a non-empty interior set, thenΘ R̦(X; ι) is a 𝓒𝓢_Θ(S; ι)-rough semigroup.

Example 4.14

According to Example 4.4, suppose that X := {s₃, s₄, s₅} is a subset of S, then we have Θ(X; 0.9) := {s₂, s₃, s₄, s₅} and Θ(X; 0.9) := {s₅}. Thus we see that Θ_bnd(X; 0.9) ≠ ∅. Hence it is straightforward to check that Θ(X; 0.9) is an 𝓒𝓢_Θ(S; 0.9)-upper approximation semigroup and Θ(X; 0.9) is a 𝓒𝓢_Θ(S; 0.9)-lower approximation semigroup. However, X is not a subsemigroup of S. Consequently, Θ R̦(X; 0.9) is a 𝓒𝓢_Θ(S; 0.9)-rough semigroup, but X is not a subsemigroup of S.

Theorem 4.15

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CPF. IfXis an ideal ofS, thenΘ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation ideal.

Proof

Suppose that X is an ideal of S. Then SX ⊆ X. From Proposition 3.11 (9), it follows that Θ(SX; ι) ⊆ Θ(X; ι). By Proposition 3.11 (1), we obtain that Θ(S; ι) = S. From Proposition 4.7, it follows that

Hence Θ(X; ι) is a left ideal of S.

Similarly, we can prove that Θ(X; ι) is a right ideal of S. Therefore we have Θ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation ideal. □

Theorem 4.16

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CF. IfXis an ideal ofSwithΘ(X; ι) ≠ ∅, thenΘ(X; ι) is a 𝓒𝓢_Θ(S; ι)-lower approximation ideal.

Proof

Suppose that X is an ideal of S. Then SX ⊆ X. From Proposition 3.11 (9), it follows that Θ(SX; ι) ⊆ Θ(X; ι). By Proposition 3.11 (1), we obtain that Θ(S; ι) = S. From Proposition 4.8, it follows that

Thus Θ(X; ι) is a left ideal of S.

Similarly, we can prove that Θ(X; ι) is a right ideal of S. Thus Θ(X; ι) is a 𝓒𝓢_Θ(S; ι)-lower approximation ideal. □

Corollary 4.17

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CF. IfXis an ideal ofSover a non-empty interior set, thenΘ R̦(X; ι) is a 𝓒𝓢_Θ(S; ι)-rough ideal.

Example 4.18

According to Example 4.4, if X := {s₁, s₃, s₅} is a subset of S, then we have Θ(X; 0.9) = S and Θ(X; 0.9) := {s₁, s₅}. Thus we see that Θ_bnd(X; 0.9) ≠ ∅. Obviously, Θ(X; 0.9) is an 𝓒𝓢_Θ(S; 0.9)-upper approximation ideal, and it is straightforward to check that Θ(X; 0.9) is a 𝓒𝓢_Θ(S; 0.9)-lower approximation ideal. However, X is not an ideal of S. Consequently, Θ R̦(X; 0.9) is a 𝓒𝓢_Θ(S; 0.9)-rough ideal, but X is not an ideal of S.

Theorem 4.19

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CF. IfXis a completely prime ideal ofS, thenΘ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation completely prime ideal.

Proof

We prove that Θ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation completely prime ideal. In fact, since X is an ideal of S, by Theorem 4.15, we have that Θ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation ideal. Let s₁, s₂ ∈ S such that s₁s₂ ∈ Θ(X; ι). Then by the Θ-complete property of 𝓒𝓢_Θ(S; ι), we get

Thus there exist s₃ ∈ CS_Θ(s₁; ι) and s₄ ∈ CS_Θ(s₂; ι) such that s₃s₄ ∈ X. Since X is a completely prime ideal, we have s₃ ∈ X or s₄ ∈ X. Hence we have CS_Θ(s₁; ι) ∩ X ≠ ∅ or CS_Θ(s₂; ι) ∩ X ≠ ∅, and so s₁ ∈ Θ(X; ι) or s₂ ∈ Θ(X; ι). Therefore Θ(X; ι) is a completely prime ideal of S. As a consequence, Θ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation completely prime ideal. □

Theorem 4.20

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CF. IfXis a completely prime ideal ofSwithΘ(X; ι) ≠ ∅, thenΘ(X; ι) is a 𝓒𝓢_Θ(S; ι)-lower approximation completely prime ideal.

Proof

Since X is an ideal of S, by Theorem 4.16, Θ(X; ι) is a 𝓒𝓢_Θ(S; ι)-lower approximation ideal. Let s₁, s₂ ∈ S such that s₁s₂ ∈ Θ(X; ι). Since Θ is complete, we have

Now, we suppose that s₁ ∉ Θ(X; ι). Then CS_Θ(s₁; ι) is not a subset of X. Thus there exists s₃ ∈ CS_Θ(s₁; ι) but s₃ ∉ X. For each s₄ ∈ CS_Θ(s₂; ι),

Whence s₃s₄ ∈ X. Since X is a completely prime ideal and s₃ ∉ X, we have s₄ ∈ X. Thus CS_Θ(s₂; ι) ⊆ X, which yields s₂ ∈ Θ(X; ι). Hence we get Θ(X; ι) is a completely prime ideal of S. Therefore Θ(X; ι) is a 𝓒𝓢_Θ(S; ι)-lower approximation completely prime ideal. □

Corollary 4.21

Let (S, 𝓒𝓢_Θ(S; ι)) be an 𝓒𝓢_Θ(S; ι)-approximation space type CF. IfXis a completely prime ideal ofSover a non-empty interior set, thenΘ R̦(X; ι) is a 𝓒𝓢_Θ(S; ι)-rough completely prime.

Example 4.22

According to Example 4.4, if X := {s₁, s₂, s₅} is a subset of S, then we have Θ(X; 0.9) = S and Θ(X; 0.9) := {s₁, s₅}. Thus we see that Θ_bnd(X; 0.9) ≠ ∅. Obviously, Θ(X; 0.9) is an 𝓒𝓢_Θ(S; 0.9)-upper approximation completely prime ideal, and it is straightforward to check that Θ(X; 0.9) is a 𝓒𝓢_Θ(S; 0.9)-lower approximation completely prime ideal. Here we can verify that X is an ideal of S, but it is not a completely prime ideal of S since s₃s₄ = s₂ ∈ X but s₃ ∉ X and s₄ ∉ X. As a consequence, Θ R̦(X; 0.9) is a 𝓒𝓢_Θ(S; 0.9)-rough completely prime ideal, but X is not a completely prime ideal of S.

Proposition 5.1

Letfbe an epimorphism fromSin (S, 𝓒𝓢_Θ(S; ι)) toTin (T, 𝓒𝓢_Ψ(T; ι)), whereΘis defined by for alls₁, s₂ ∈ S, Θ(s₁, s₂) = Ψ(f(s₁), f(s₂)). Then the following statements hold.

1. For alls₁, s₂ ∈ S, s₁ ∈ CS_Θ(s₂; ι) if and only if f(s₁) ∈ CS_Ψ(f(s₂); ι).

2. f(Θ(X; ι)) = Ψ(f(X); ι) for every non-empty subsetXofS.

3. f(Θ(X; ι)) ⊆ Ψ(f(X); ι) for every non-empty subsetXofS.

4. Iffis injective, then f(Θ(X; ι)) = Ψ(f(X); ι) for every non-empty subsetXofS.

5. IfΨis a compatible preorder fuzzy relation, thenΘis a compatible preorder fuzzy relation.

Proof

1. Let s₁, s₂ ∈ S be such that s₁ ∈ CS_Θ(s₂; ι). Then f(s₁), f(s₂) ∈ T and S_Θ(s₁; ι) = S_Θ(s₂; ι). In the following, we shall prove that S_Ψ(f(s₁); ι) = S_Ψ(f(s₂); ι). Let t₁ ∈ S_Ψ(f(s₁); ι). Then Ψ(f(s₁), t₁) ≥ ι. Since f is surjective, there exists s₃ ∈ S such that f(s₃) = t₁. Whence Ψ(f(s₁), f(s₃)) ≥ ι, and so Θ(s₁, s₃) ≥ ι. Thus s₃ ∈ S_Θ(s₁; ι). Whence we have s₃ ∈ S_Θ(s₂; ι). Hence Θ(s₂, s₃) ≥ ι, and so Ψ(f(s₂), f(s₃)) ≥ ι. Thus t₁ = f(s₃) ∈ S_Ψ(f(s₂); ι). Then we have S_Ψ(f(s₁); ι) ⊆ S_Ψ(f(s₂); ι). Similarly, we can show that S_Ψ(f(s₂); ι) ⊆ S_Ψ(f(s₁); ι). Therefore S_Ψ(f(s₁); ι) = S_Ψ(f(s₂); ι). As a consequence, f(s₁) ∈ CS_Ψ(f(s₂); ι).

   Conversely, it is easy to verify that s₁ ∈ CS_Θ(s₂; ι) whenever f(s₁) ∈ CS_Ψ(f(s₂); ι) for all s₁, s₂ ∈ S.

2. Let X be a non-empty subset of S. We verify firstly that f(Θ(X; ι)) = Ψ(f(X); ι). Suppose that t₁ ∈ f(Θ(X; ι)). Then there exists s₁ ∈ Θ(X; ι) such that f(s₁) = t₁. Therefore we have CS_Θ(s₁; ι) ∩ X ≠ ∅. Thus there exists s₂ ∈ S such that s₂ ∈ CS_Θ(s₁; ι) and s₂ ∈ X. By the argument (1), we obtain that f(s₂) ∈ CS_Ψ(f(s₁); ι) and f(s₂) ∈ f(X). Then we have CS_Ψ(f(s₁); ι) ∩ f(X) ≠ ∅, and so t₁ = f(s₁) ∈ Ψ(f(X); ι). Thus we have f(Θ(X; ι)) ⊆ Ψ(f(X); ι).

   On the other hand, let t₂ ∈ Ψ(f(X); ι). Then there exists s₃ ∈ S such that f(s₃) = t₂, and so CS_Ψ(f(s₃); ι) ∩ f(X) ≠ ∅. Thus there exists s₄ ∈ X such that f(s₄) ∈ f(X) and f(s₄) ∈ CS_Ψ(f(s₃); ι). By the argument (1), we get that s₄ ∈ CS_Θ(s₃; ι), and so we have CS_Θ(s₃; ι) ∩ X ≠ ∅. Hence s₃ ∈ Θ(X; ι), and so t₂ = f(s₃) ∈ f(Θ(X; ι)). Thus we get Ψ(f(X); ι) ⊆ f(Θ(X; ι)). This implies that f(Θ(X; ι)) = Ψ(f(X); ι).

3. Let X be a non-empty subset of S. Let t₁ ∈ f(Θ(X; ι)). Then there exists s₁ ∈ Θ(X; ι) such that f(s₁) = t₁. Thus we get CS_Θ(s₁; ι) ⊆ X. We shall prove that CS_Ψ(t₁; ι) ⊆ f(X). Let t₂ ∈ CS_Ψ(t₁; ι). Then there exist s₂ ∈ S such that f(s₂) = t₂. Thus we have f(s₂) ∈ CS_Ψ(f(s₁); ι). By the argument (1), we obtain that s₂ ∈ CS_Θ(s₁; ι), and so s₂ ∈ X. Hence we have t₂ = f(s₂) ∈ f(X), and Thus CS_Ψ(t₁; ι) ⊆ f(X). Therefore we have t₁ ∈ Ψ(f(X); ι). As a consequence, f(Θ(X; ι)) ⊆ Ψ(f(X); ι).

4. Let X be a non-empty subset of S. We only need to prove that Ψ(f(X); ι) ⊆ f(Θ(X; ι)). Suppose that t₁ ∈ Ψ(f(X); ι). Then there exists s₁ ∈ S such that f(s₁) = t₁. Thus we have CS_Ψ(f(s₁); ι) ⊆ f(X). We shall show that CS_Θ(s₁; ι) ⊆ X. Let s₂ ∈ CS_Θ(s₁; ι). Then by the argument (1), we have f(s₂) ∈ CS_Ψ(f(s₁); ι). Hence f(s₂) ∈ f(X). Thus there exists s₃ ∈ X such that f(s₃) = f(s₂). By the assumption, we have s₂ ∈ X, and so CS_Θ(s₁; ι) ⊆ X. Hence s₁ ∈ Θ(X; ι), and so t₁ = f(s₁) ∈ f(Θ(X; ι)). Thus Ψ(f(X); ι) ⊆ f(Θ(X; ι)).

   By the argument (3), we get f(Θ(X; ι)) ⊆ Ψ(f(X); ι). Consequently, f(Θ(X; ι)) = Ψ(f(X); ι).

5. The proof is straightforward, so we omit it. □

Proposition 5.2

Letfbe an isomorphism fromSin (S, 𝓒𝓢_Θ(S; ι)) toTin (T, 𝓒𝓢_Ψ(T; ι)), whereΘis defined by for alls₁, s₂ ∈ S, Θ(s₁, s₂) = Ψ(f(s₁), f(s₂)). IfΨis complete, thenΘis complete.

Proof

Let s₁, s₂ be two elements in S and let s₃ ∈ CS_Θ(s₁s₂; ι). Then by Proposition 5.1 (1), we get that f(s₃) ∈ CS_Ψ(f(s₁s₂); ι). Since f is a homomorphism and Ψ is complete, we have

Thus there exist t₁ ∈ CS_Ψ(f(s₁); ι) and t₂ ∈ CS_Ψ(f(s₂); ι) such that f(s₃) = t₁t₂. Since f is surjective, there exist s₄, s₅ ∈ S such that f(s₄) = t₁ and f(s₅) = t₂. From

it follows that f(s₄) ∈ CS_Ψ(f(s₁); ι) and f(s₅) ∈ CS_Ψ(f(s₂); ι). By Proposition 5.1 (1), we obtain that s₄ ∈ CS_Θ(s₁; ι) and s₅ ∈ CS_Θ(s₂; ι). Since f is a homomorphism, we have f(s₃) = f(s₄)f(s₅) = f(s₄s₅). Since f is injective, we get s₃ = s₄s₅. Thus we get that s₃ ∈ CS_Θ(s₁; ι)CS_Θ(s₂; ι). Therefore we have CS_Θ(s₁s₂; ι) ⊆ CS_Θ(s₁; ι)CS_Θ(s₂; ι).

On the other hand, by Proposition 4.2 and Proposition 5.1 (5), CS_Θ(s₁; ι)CS_Θ(s₂; ι) ⊆ CS_Θ(s₁s₂; ι). Thus CS_Θ(s₁; ι)CS_Θ(s₂; ι) = CS_Θ(s₁s₂; ι). Hence 𝓒𝓢_Θ(S; ι) is Θ-complete. Therefore Θ is complete. □

Theorem 5.3

Letfbe an epimorphism fromSin (S, 𝓒𝓢_Θ(S; ι)) toTin (T, 𝓒𝓢_Ψ(T; ι)) type CPF, whereΘis defined by for alls₁, s₂ ∈ S, Θ(s₁, s₂) = Ψ(f(s₁), f(s₂)). IfXis a non-empty subset ofS, thenΘ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation semigroup if and only ifΨ(f(X); ι) is an 𝓒𝓢_Ψ(T; ι)-upper approximation semigroup.

Proof

Suppose that Θ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation semigroup. Then by Proposition 5.1 (2),

Hence Ψ(f(X); ι) is a subsemigroup of T. Thus Ψ(f(X); ι) is an 𝓒𝓢_Ψ(T; ι)-upper approximation semigroup.

Conversely, let s₁ ∈ (Θ(X; ι))(Θ(X; ι)). From Proposition 5.1 (2), it follows that

Thus there exists s₂ ∈ Θ(X; ι) such that f(s₁) = f(s₂). Hence we have CS_Θ(s₂; ι) ∩ X ≠ ∅. From Proposition 3.4 (1), it follows that f(s₁) ∈ CS_Ψ(f(s₂); ι). By Proposition 5.1 (1), we obtain that s₁ ∈ CS_Θ(s₂; ι). From Proposition 3.4 (2), it follows that CS_Θ(s₁; ι) = CS_Θ(s₂; ι). Thus we have CS_Θ(s₁; ι) ∩ X ≠ ∅, and so s₁ ∈ Θ(X; ι). Hence we have that (Θ(X; ι))(Θ(X; ι)) ⊆ Θ(X; ι). Thus Θ(X; ι) is a subsemigroup of S. Therefore Θ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation semigroup. □

Theorem 5.4

Letfbe an isomorphism fromSin (S, 𝓒𝓢_Θ(S; ι)) toTin (T, 𝓒𝓢_Ψ(T; ι)) type CPF, whereΘis defined by for alls₁, s₂ ∈ S, Θ(s₁, s₂) = Ψ(f(s₁), f(s₂)). IfXis a non-empty subset ofS, thenΘ(X; ι) is a 𝓒𝓢_Θ(S; ι)-lower approximation semigroup if and only ifΨ(f(X); ι) is a 𝓒𝓢_Ψ(T; ι)-lower approximation semigroup.

Proof

By Proposition 5.1 (4) and using the similar method in the proof of Theorem 5.3, we can prove that the statement holds. □

Corollary 5.5

Letfbe an isomorphism fromSin (S, 𝓒𝓢_Θ(S; ι)) toTin (T, 𝓒𝓢_Ψ(T; ι)) type CPF, whereΘis defined by for alls₁, s₂ ∈ S, Θ(s₁, s₂) = Ψ(f(s₁), f(s₂)). IfXis a non-empty subset ofS, thenΘ R̦(X; ι) is a 𝓒𝓢_Θ(S; ι)-rough semigroup if and only ifΨR̦(f(X); ι) is a 𝓒𝓢_Ψ(T; ι)-rough semigroup.

Theorem 5.6

Letfbe an epimorphism fromSin (S, 𝓒𝓢_Θ(S; ι)) toTin (T, 𝓒𝓢_Ψ(T; ι)) type CPF, whereΘis defined by for alls₁, s₂ ∈ S, Θ(s₁, s₂) = Ψ(f(s₁), f(s₂)). IfXis a non-empty subset ofS, thenΘ(X; ι) is an 𝓒𝓢_Θ(S; ι)-upper approximation ideal if and only ifΨ(f(X); ι) is an 𝓒𝓢_Ψ(T; ι)-upper approximation ideal.