In this section, we propose the definition of compatible preorder fuzzy relations on semigroups. Then we introduce the roughness in semigroups induced by compatible preorder fuzzy relations. We provide sufficient conditions for them and give some interesting properties and examples.

#### 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*

$$\begin{array}{}{\displaystyle (C{S}_{\mathit{\Theta}}({s}_{1};\iota ))(C{S}_{\mathit{\Theta}}({s}_{2};\iota ))\subseteq C{S}_{\mathit{\Theta}}({s}_{1}{s}_{2};\iota )}\end{array}$$

*for all* *s*_{1}, *s*_{2} ∈ *S*.

#### Proof

Let *s*_{1}, *s*_{2} be two elements in *S* and let *s*_{3} ∈ (*CS*_{Θ}(*s*_{1}; *ι*)) (*CS*_{Θ}(*s*_{2}; *ι*)). Then there exist *s*_{4} ∈ *CS*_{Θ}(*s*_{1}; *ι*) and *s*_{5} ∈ *CS*_{Θ}(*s*_{2}; *ι*) such that *s*_{3} = *s*_{4}*s*_{5}. Thus *S*_{Θ}(*s*_{1}; *ι*) = *S*_{Θ}(*s*_{4}; *ι*) and *S*_{Θ}(*s*_{2}; *ι*) = *S*_{Θ}(*s*_{5}; *ι*). Hence we get that *S*_{Θ}(*s*_{1}*s*_{2}; *ι*) = *S*_{Θ}(*s*_{4}*s*_{5}; *ι*). Indeed, we suppose that *s*_{6} ∈ *S*_{Θ}(*s*_{4}*s*_{5}; *ι*). Then we have *Θ*(*s*_{4}*s*_{5}, *s*_{6}) ≥ *ι*. Since *Θ* is reflexive, we have *Θ*(*s*_{4}, *s*_{4}) = *Θ*(*s*_{5}, *s*_{5}) = 1 ≥ *ι*, and so *s*_{4} ∈ *S*_{Θ}(*s*_{4}; *ι*) and *s*_{5} ∈ *S*_{Θ}(*s*_{5}; *ι*). Whence *s*_{4} ∈ *S*_{Θ}(*s*_{1}; *ι*) and *s*_{5} ∈ *S*_{Θ}(*s*_{2}; *ι*). Thus *Θ*(*s*_{1}, *s*_{4}) ≥ *ι* and *Θ*(*s*_{2}, *s*_{5}) ≥ *ι*. Since *Θ* is transitive and compatible, we have

$$\begin{array}{}{\displaystyle \mathit{\Theta}({s}_{1}{s}_{2},{s}_{4}{s}_{5})\ge {\vee}_{{s}_{7}\in S}(\mathit{\Theta}({s}_{1}{s}_{2},{s}_{7})\wedge \mathit{\Theta}({s}_{7},{s}_{4}{s}_{5}))}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\ge \phantom{\rule{thickmathspace}{0ex}}\mathit{\Theta}({s}_{1}{s}_{2},{s}_{4}{s}_{2})\wedge \mathit{\Theta}({s}_{4}{s}_{2},{s}_{4}{s}_{5})}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\ge \phantom{\rule{thickmathspace}{0ex}}\mathit{\Theta}({s}_{1},{s}_{4})\wedge \mathit{\Theta}({s}_{2},{s}_{5})}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\ge \phantom{\rule{thickmathspace}{0ex}}\iota \wedge \iota}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\iota .}\end{array}$$

Hence *Θ*(*s*_{1}*s*_{2}, *s*_{4}*s*_{5}) ≥ *ι*. Since *Θ* is transitive, we have

$$\begin{array}{}{\displaystyle \mathit{\Theta}({s}_{1}{s}_{2},{s}_{6})\ge {\vee}_{{s}_{8}\in S}(\mathit{\Theta}({s}_{1}{s}_{2},{s}_{8})\wedge \mathit{\Theta}({s}_{8},{s}_{6}))}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\ge \phantom{\rule{thickmathspace}{0ex}}\mathit{\Theta}({s}_{1}{s}_{2},{s}_{4}{s}_{5})\wedge \mathit{\Theta}({s}_{4}{s}_{5},{s}_{6})}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\ge \phantom{\rule{thickmathspace}{0ex}}\iota \wedge \iota}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}=\phantom{\rule{thickmathspace}{0ex}}\iota .}\end{array}$$

Thus *Θ*(*s*_{1}*s*_{2}, *s*_{6}) ≥ *ι*, and so *s*_{6} ∈ *S*_{Θ}(*s*_{1}*s*_{2}; *ι*). Hence *S*_{Θ}(*s*_{4}*s*_{5}; *ι*) ⊆ *S*_{Θ}(*s*_{1}*s*_{2}; *ι*). Similarly, we can show that *S*_{Θ}(*s*_{1}*s*_{2}; *ι*) ⊆ *S*_{Θ}(*s*_{4}*s*_{5}; *ι*). Thus *S*_{Θ}(*s*_{1}*s*_{2}; *ι*) = *S*_{Θ}(*s*_{4}*s*_{5}; *ι*), which yields *s*_{3} ∈ *CS*_{Θ}(*s*_{1}*s*_{2}; *ι*). This implies that (*CS*_{Θ}(*s*_{1}; *ι*)) (*CS*_{Θ}(*s*_{2}; *ι*)) ⊆ *CS*_{Θ}(*s*_{1}*s*_{2}; *ι*). □

In the following, we give an example to illustrate that the property in Proposition 4.2 is indispensable.

#### Example 4.3

Let *S* := {*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}, *s*_{5}} be a semigroup with multiplication rules defined by .

Table 1 The multiplication table on *S*

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

$$\begin{array}{}{\displaystyle \left(\begin{array}{ccccc}1& 0& 1& 0& 1\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 1\\ 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 1\end{array}\right)}\end{array}$$

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}(*s*_{1}; 0.9) := {*s*_{1}, *s*_{3}, *s*_{5}},

*S*_{Θ}(*s*_{2}; 0.9) := {*s*_{2}},

*S*_{Θ}(*s*_{3}; 0.9) := {*s*_{3}, *s*_{5}},

*S*_{Θ}(*s*_{4}; 0.9) := {*s*_{4}} and

*S*_{Θ}(*s*_{5}; 0.9) := {*s*_{3}, *s*_{5}}.

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

*CS*_{Θ}(*s*_{1}; 0.9) := {*s*_{1}},

*CS*_{Θ}(*s*_{2}; 0.9) := {*s*_{2}},

*CS*_{Θ}(*s*_{3}; 0.9) := {*s*_{3}, *s*_{5}},

*CS*_{Θ}(*s*_{4}; 0.9) := {*s*_{4}} and

*CS*_{Θ}(*s*_{5}; 0.9) := {*s*_{3}, *s*_{5}}.

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*.

Observe that, in Example 4.3, it does not hold in general for the equality case. Now, we consider the following example.

#### Example 4.4

Let *S* := {*s*_{1}, *s*_{2}, *s*_{3}, *s*_{4}, *s*_{5}} be a semigroup with multiplication rules defined by .

Table 2 The multiplication table on *S*

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

$$\begin{array}{}{\displaystyle \left(\begin{array}{ccccc}1& 0& 0& 0& 1\\ 0& 1& 1& 1& 0\\ 0& 1& 1& 1& 0\\ 0& 1& 1& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right)}\end{array}$$

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}(*s*_{1}; 0.9) := {*s*_{1}, *s*_{5}},

*S*_{Θ}(*s*_{2}; 0.9) := {*s*_{2}, *s*_{3}, *s*_{4}},

*S*_{Θ}(*s*_{3}; 0.9) := {*s*_{2}, *s*_{3}, *s*_{4}},

*S*_{Θ}(*s*_{4}; 0.9) := {*s*_{2}, *s*_{3}, *s*_{4}} and

*S*_{Θ}(*s*_{5}; 0.9) := {*s*_{5}}.

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

*CS*_{Θ}(*s*_{1}; 0.9) := {*s*_{1}},

*CS*_{Θ}(*s*_{2}; 0.9) := {*s*_{2}, *s*_{3}, *s*_{4}},

*CS*_{Θ}(*s*_{3}; 0.9) := {*s*_{2}, *s*_{3}, *s*_{4}},

*CS*_{Θ}(*s*_{4}; 0.9) := {*s*_{2}, *s*_{3}, *s*_{4}} and

*CS*_{Θ}(*s*_{5}; 0.9) := {*s*_{5}}.

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 by* *Θ* (briefly, *Θ*-*complete*) if for all *s*_{1}, *s*_{2} ∈ *S*,

$$\begin{array}{}{\displaystyle (C{S}_{\mathit{\Theta}}({s}_{1};\iota ))(C{S}_{\mathit{\Theta}}({s}_{2};\iota ))=C{S}_{\mathit{\Theta}}({s}_{1}{s}_{2};\iota ).}\end{array}$$

#### 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*

$$\begin{array}{}{\displaystyle (\overline{\mathit{\Theta}}(X;\iota ))(\overline{\mathit{\Theta}}(Y;\iota ))\subseteq \overline{\mathit{\Theta}}(XY;\iota ),}\end{array}$$

*for every non*-*empty subsets X*, *Y* *of* *S*.

#### Proof

Let *X* and *Y* be two non-empty subsets of *S*. Suppose that *s*_{1} ∈ (*Θ*(*X*; *ι*))(*Θ*(*Y*; *ι*)). Then there exist *s*_{2} ∈ *Θ*(*X*; *ι*) and *s*_{3} ∈ *Θ*(*Y*; *ι*) such that *s*_{1} = *s*_{2}*s*_{3}. Thus we have that *CS*_{Θ}(*s*_{2}; *ι*) ∩ *X* ≠ ∅ and *CS*_{Θ}(*s*_{3}; *ι*) ∩ *Y* ≠ ∅. Then there exist *s*_{4}, *s*_{5} ∈ *S* such that *s*_{4} ∈ *CS*_{Θ}(*s*_{2}; *ι*) ∩ *X* and *s*_{5} ∈ *CS*_{Θ}(*s*_{3}; *ι*) ∩ *Y*. From Proposition 4.2, it follows that *s*_{4}*s*_{5} ∈ (*CS*_{Θ}(*s*_{2}; *ι*))(*CS*_{Θ}(*s*_{3}; *ι*)) ⊆ *CS*_{Θ}(*s*_{2}*s*_{3}; *ι*) and *s*_{4}*s*_{5} ∈ *XY*. Thus *CS*_{Θ}(*s*_{2}*s*_{3}; *ι*) ∩ *XY* ≠ ∅, which yields *s*_{1} = *s*_{2}*s*_{3} ∈ *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*

$$\begin{array}{}{\displaystyle (\underset{\_}{\mathit{\Theta}}(X;\iota ))(\underset{\_}{\mathit{\Theta}}(Y;\iota ))\subseteq \underset{\_}{\mathit{\Theta}}(XY;\iota ),}\end{array}$$

*for every non*-*empty subsets X*, *Y* *of* *S*.

#### Proof

Let *X* and *Y* be two non-empty subsets of *S* and let *s*_{1} ∈ (*Θ*(*X*; *ι*))(*Θ*(*Y*; *ι*)). Then there exist *s*_{2} ∈ *Θ*(*X*; *ι*) and *s*_{3} ∈ *Θ*(*Y*; *ι*) such that *s*_{1} = *s*_{2}*s*_{3}, and so *CS*_{Θ}(*s*_{2}; *ι*) ⊆ *X* and *CS*_{Θ}(*s*_{3}; *ι*) ⊆ *Y*. Since *Θ* is complete, we get *CS*_{Θ}(*s*_{2}*s*_{3}; *ι*) = *CS*_{Θ}(*s*_{2}; *ι*)*CS*_{Θ}(*s*_{3}; *ι*) ⊆ *XY*. Thus *CS*_{Θ}(*s*_{2}*s*_{3}; *ι*) ⊆ *XY*. Hence *s*_{1} = *s*_{2}*s*_{3} ∈ *Θ*(*XY*; *ι*). Therefore (*Θ*(*X*; *ι*))(*Θ*(*Y*; *ι*)) ⊆ *Θ*(*XY*; *ι*). □

We consider the following example.

#### Example 4.9

According to Example 4.4, suppose that *X* := {*s*_{1}, *s*_{4}, *s*_{5}} is a subset of *S*. Then we have *Θ*(*X*; *ι*) = *S* and *Θ*(*X*; *ι*) := {*s*_{1}, *s*_{5}}. 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*. *If* *X* *is a subsemigroup of* *S*, *then* *Θ*(*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

$$\begin{array}{}{\displaystyle (\overline{\mathit{\Theta}}(X;\iota ))(\overline{\mathit{\Theta}}(X;\iota ))\subseteq \overline{\mathit{\Theta}}(XX;\iota )\subseteq \overline{\mathit{\Theta}}(X;\iota ).}\end{array}$$

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*. *If* *X* *is a subsemigroup of* *S* *with* *Θ*(*X*; *ι*) ≠ ∅, *then* *Θ*(*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

$$\begin{array}{}{\displaystyle (\underset{\_}{\mathit{\Theta}}(X;\iota ))(\underset{\_}{\mathit{\Theta}}(X;\iota ))\subseteq \underset{\_}{\mathit{\Theta}}(XX;\iota )\subseteq \underset{\_}{\mathit{\Theta}}(X;\iota ).}\end{array}$$

Thus *Θ*(*X*; *ι*) is a subsemigroup of *S*. Therefore *Θ*(*X*; *ι*) is a 𝓒𝓢_{Θ}(*S*; *ι*)-lower approximation semigroup. □

The following corollary is an immediate consequence of Proposition 3.13, Theorem 4.11 and Theorem 4.12.

#### Corollary 4.13

*Let* (*S*, 𝓒𝓢_{Θ}(*S*; *ι*)) *be an* 𝓒𝓢_{Θ}(*S*; *ι*)-*approximation space type CF*. *If* *X* *is a subsemigroup of* *S* *over a non*-*empty interior set*, *then* *Θ R̦*(*X*; *ι*) *is a* 𝓒𝓢_{Θ}(*S*; *ι*)-*rough semigroup*.

Observe that, in Corollary 4.13, the converse is not true in general. We present an example as the following.

#### Example 4.14

According to Example 4.4, suppose that *X* := {*s*_{3}, *s*_{4}, *s*_{5}} is a subset of *S*, then we have *Θ*(*X*; 0.9) := {*s*_{2}, *s*_{3}, *s*_{4}, *s*_{5}} and *Θ*(*X*; 0.9) := {*s*_{5}}. 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*. *If* *X* *is an ideal of* *S*, *then* *Θ*(*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

$$\begin{array}{}{\displaystyle S(\overline{\mathit{\Theta}}(X;\iota ))=(\overline{\mathit{\Theta}}(S;\iota ))(\overline{\mathit{\Theta}}(X;\iota ))\subseteq \overline{\mathit{\Theta}}(SX;\iota )\subseteq \overline{\mathit{\Theta}}(X;\iota ).}\end{array}$$

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*. *If* *X* *is an ideal of* *S* *with* *Θ*(*X*; *ι*) ≠ ∅, *then* *Θ*(*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

$$\begin{array}{}{\displaystyle S(\underset{\_}{\mathit{\Theta}}(X;\iota ))=(\underset{\_}{\mathit{\Theta}}(S;\iota ))(\underset{\_}{\mathit{\Theta}}(X;\iota ))\subseteq \underset{\_}{\mathit{\Theta}}(SX;\iota )\subseteq \underset{\_}{\mathit{\Theta}}(X;\iota ).}\end{array}$$

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. □

The following corollary is an immediate consequence of Proposition 3.13, Theorem 4.15 and Theorem 4.16.

#### Corollary 4.17

*Let* (*S*, 𝓒𝓢_{Θ}(*S*; *ι*)) *be an* 𝓒𝓢_{Θ}(*S*; *ι*)-*approximation space type CF*. *If* *X* *is an ideal of* *S* *over a non*-*empty interior set*, *then* *Θ R̦*(*X*; *ι*) *is a* 𝓒𝓢_{Θ}(*S*; *ι*)-*rough ideal*.

Observe that, in Corollary 4.17, the converse is not true in general. We present an example as the following.

#### Example 4.18

According to Example 4.4, if *X* := {*s*_{1}, *s*_{3}, *s*_{5}} is a subset of *S*, then we have *Θ*(*X*; 0.9) = *S* and *Θ*(*X*; 0.9) := {*s*_{1}, *s*_{5}}. 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*. *If* *X* *is a completely prime ideal of* *S*, *then* *Θ*(*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 *s*_{1}, *s*_{2} ∈ *S* such that *s*_{1}*s*_{2} ∈ *Θ*(*X*; *ι*). Then by the *Θ*-complete property of 𝓒𝓢_{Θ}(*S*; *ι*), we get

$$\begin{array}{}{\displaystyle (C{S}_{\mathit{\Theta}}({s}_{1};\iota ))(C{S}_{\mathit{\Theta}}({s}_{2};\iota ))\cap X=C{S}_{\mathit{\Theta}}({s}_{1}{s}_{2};\iota )\cap X\ne \mathrm{\varnothing}.}\end{array}$$

Thus there exist *s*_{3} ∈ *CS*_{Θ}(*s*_{1}; *ι*) and *s*_{4} ∈ *CS*_{Θ}(*s*_{2}; *ι*) such that *s*_{3}*s*_{4} ∈ *X*. Since *X* is a completely prime ideal, we have *s*_{3} ∈ *X* or *s*_{4} ∈ *X*. Hence we have *CS*_{Θ}(*s*_{1}; *ι*) ∩ *X* ≠ ∅ or *CS*_{Θ}(*s*_{2}; *ι*) ∩ *X* ≠ ∅, and so *s*_{1} ∈ *Θ*(*X*; *ι*) or *s*_{2} ∈ *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*. *If* *X* *is a completely prime ideal of* *S* *with* *Θ*(*X*; *ι*) ≠ ∅, *then* *Θ*(*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 *s*_{1}, *s*_{2} ∈ *S* such that *s*_{1}*s*_{2} ∈ *Θ*(*X*; *ι*). Since *Θ* is complete, we have

$$\begin{array}{}{\displaystyle (C{S}_{\mathit{\Theta}}({s}_{1};\iota ))(C{S}_{\mathit{\Theta}}({s}_{2};\iota ))=C{S}_{\mathit{\Theta}}({s}_{1}{s}_{2};\iota )\subseteq X.}\end{array}$$

Now, we suppose that *s*_{1} ∉ *Θ*(*X*; *ι*). Then *CS*_{Θ}(*s*_{1}; *ι*) is not a subset of *X*. Thus there exists *s*_{3} ∈ *CS*_{Θ}(*s*_{1}; *ι*) but *s*_{3} ∉ *X*. For each *s*_{4} ∈ *CS*_{Θ}(*s*_{2}; *ι*),

$$\begin{array}{}{\displaystyle {s}_{3}{s}_{4}\in (C{S}_{\mathit{\Theta}}({s}_{1};\iota ))(C{S}_{\mathit{\Theta}}({s}_{2};\iota ))\subseteq X.}\end{array}$$

Whence *s*_{3}*s*_{4} ∈ *X*. Since *X* is a completely prime ideal and *s*_{3} ∉ *X*, we have *s*_{4} ∈ *X*. Thus *CS*_{Θ}(*s*_{2}; *ι*) ⊆ *X*, which yields *s*_{2} ∈ *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. □

The following corollary is an immediate consequence of Proposition 3.13, Theorem 4.19 and Theorem 4.20.

#### Corollary 4.21

*Let* (*S*, 𝓒𝓢_{Θ}(*S*; *ι*)) *be an* 𝓒𝓢_{Θ}(*S*; *ι*)-*approximation space type CF*. *If* *X* *is a completely prime ideal of* *S* *over a non*-*empty interior set*, *then* *Θ R̦*(*X*; *ι*) *is a* 𝓒𝓢_{Θ}(*S*; *ι*)-*rough completely prime*.

Observe that, in Corollary 4.21, the converse is not true in general. We present an example as the following.

#### Example 4.22

According to Example 4.4, if *X* := {*s*_{1}, *s*_{2}, *s*_{5}} is a subset of *S*, then we have *Θ*(*X*; 0.9) = *S* and *Θ*(*X*; 0.9) := {*s*_{1}, *s*_{5}}. 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 *s*_{3}*s*_{4} = *s*_{2} ∈ *X* but *s*_{3} ∉ *X* and *s*_{4} ∉ *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*.

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