In the section, we introduce a new congruence relation of semigroups. Referring to [18], we study rough semigroups and, at the same time, we provide some examples. Throughout this paper, *S* is a semigroup.

*Let μ be a fuzzy ideal of S. For each t* ∈ [0, *μ*(*0*)], *the set U*(*μ*, *t*) = {(*x*, *y*) ∈ *× S*|(*μ*(*x*) ∧ *μ*(*y*)) ∨ *I d*_{S}(*x*, *y*) ≥ *t*} *is called a t-level relation of μ*.

For *I ds*(*x*, *y*), we know *x* = *y*, then *I ds*(*x*, *y*) = 1; *x* ≠ *y*, then *I ds*(*x*, *y*) = 0. Next, we prove *U*(*μ*, *t*)is a congruence relation.

*Let μ be a fuzzy ideal of S and t* ∈ [*0*, *μ*(*0*)].*Then U*(*μ*, *t*) *is a congruence relation on S*.

First of all, we show *U*(*μ*, *t*) is an equivalence relation.

Reflexive: For any element *x* ∈ *S*. (*μ*(*x*)∧ *μ*(*x*))∨ *I ds*(*x*, *x*) = *I ds*(*x*, *x*) = 1 ≥ *t*.

Symmetry: Obviously, *U*(*μ*, *t*) is symmetric.

Transitivity: Let (*x*, *y*) ∈ *U*(*μ*, *t*) and (*y*, *z*) ∈ *U*(*μ*, *t*), and then we have (*μ*(*x*) ∧ *μ*(*y*)) ∨ *I ds*(*x*, *y*) ≥ *t*,(*μ* (*y*) ∧ μ (*z*))∨ *I ds*(*y*, *z*) ≥ *t*. If *x* = *y* = *z*, it is clear that (*x*, *z*) ∈ *U*(*μ*, *t*); If *x* = *y* ≠ *z*, then *μ*(*y*, *z*) ≥ *t* and ((*μ*(*x*) ∧ *μ*(*z*))∨ *I ds*(*x*, *z*) = *μ* (*x*) ∧ *μ*(*z*) = *μ* (*y*) ∧ *μ* (*z*) ≥ *t*, therefore, (*x*, *z*) ∈ *U*(*μ*, *t*); if *x* ≠ *y* = *z*, we have *μ*(*x*) ∧ *μ* (*y*) ≥ *t* and (*μ* (*x*) ∧ *μ*(*z*))∨ *I ds*(*x*, *z*) = *μ*(*x*) ∧ *μ*(*z*) = *μ* (*x*) ∧ *μ* (*y*) ≥ *t*, so (*x*, *z*) ∈ *U*(*μ*, *t*); if *x* ≠ *y* ≠ *z*, then we have *μ*(*x*) ∧ *μ*(*y*) ≥ *t*, *μ*(*y*) ∧ *μ* (*z*) ≥ *t* and (*μ*(*x*) ∧ *μ*(*z*))∨ *I ds*(*x*, *z*) = *μ* (*x*) ∧ *μ*(*z*) ≥ *μ*(*x*) ∧ *μ*(*y*) ∧ *μ*(*z*) ≥ *t*, so (*x*, *z*) ∈ *U*(*μ*, *t*) . Conclusion, *U*(*μ*, *t*) is an equivalence relation.

Next, we prove that *U*(*μ*, *t*) is a congruence relation. For (*x*, *y*) ∈ *U*(*μ*, *t*), we prove (*ax*, *ay*) ∈ *U*(*μ*, *t*) and (*xa*, *ya*) ∈ *U*(*μ*, *t*) is also right. In the following, we only prove the former, the latter is the same. In other words, we only prove (*μ* (*ax*) ∧ *μ*(*ay*))∨ *I ds*(*ax*, *ay*) ≥ *t*. If *ax* = *ay*, clearly, 1 ≥ *t*. If *ax* ≠ *ay*, then *x* ≠ *y*, then
$$\begin{array}{}(\mu (ax)\wedge \mu (ay))\vee Ids(ax,ay)=\mu (ax)\wedge \mu (ay)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\underset{\_}{>}[\mu (a)\vee \mu (x)]\wedge [\mu (a)\vee \mu (y)]\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}=[\mu (a)\wedge \mu (a)]\vee [\mu (x)\wedge \mu (a)]\vee [\mu (a)\wedge \mu (y)]\vee [\mu (x)\wedge \mu (y)]\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\underset{\_}{>}\mu (x)\wedge \mu (y)\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\underset{\_}{>}t\end{array}$$

Thus, *U*(*μ*, *t*) is a congruence relation.

For any fuzzy ideal *μ* of *S*, we know that *μ*(0) ≥ *μ*(*x*) and *μ*(0) ≤ 1, so when *t* ∈ [0, *μ*(0)], the above lemma is proper. We say *x* is congruent to *y* model *μ*, written *x* ≡_{t} *y*(*mod μ*). If for elements *x*, *y* ∈ *S*, *t* ∈ [0,1], (*μ*(*x*) ∧ *μ*(*y*)) *∨ I ds*(*x*, *y*) ≥ *t*, we use [*x*]_{(μ,t)} as the equivalence class of *x*. However, *U*(*μ*, *t*) is not a complete congruence relation. Through our research, we can obtain Lemma 3.3 as follows.

*Let* *μ* *be a fuzzy ideal of* *S* *and* *t* ∈ [0, 1], *then* [*x*]_{(μ, t)}[*y*]_{(μ, t)} ⊆ [*xy*]_{(μ, t)}.

*Let* *S* = {*a*, *b*, *c*, * d*} *be a semigroup with the following* “*” *table*.

*Assume that*
$\mu =\frac{0.3}{a}+\frac{0.5}{b}+\frac{0.1}{c}+\frac{0.8}{d}$ *is a fuzzy ideal of S. Here* *t* = 0.4, *then we have* *U*(*μ*; 0.4) = {(*a*, *a*),(*b*,*b*),(*c*,*c*),(*d*,*d*),(*b*,*d*)}, *so we have* [*a*]_{(μ, 0.4)} = {*a*}, [*b*]_{(μ, 0.4)} = {*b, d*}, [*c*]_{(μ, 0.4)} = {*c*}, [*d*]_{(μ, 0.4)} = {*b, d*}. [*a*]_{(μ, 0.4)}[*c*]_{(μ, 0.4)} = {*b*}. *Here*, *a* * *c* = *b*, *so* [*a***c*]_{(μ, 0.4)} = {(*b*, *d*)}. *Obviously* [*a*]_{(μ, 0.4)}[*c*]_{(μ, 0.4)} ⊆ [*a***c*]_{(μ, 0.4)}.

From the above example, we see we can not write [*x*]_{(μ, t)}[*y*]_{(μ, t)} = [*xy*]_{(μ, t)}, it is only a containment relation. However there exists special *U*(*μ, t*) such that [*x*]_{(μ, t)}[*y*]_{(μ, t)} = [*xy*]_{(μ, t)}. So we give the definition to make *U*(*μ, t*) a complete congruence relation as follows.

*U*(*μ, t*) *is called a complete congruence relation if it satisfies*: *for any* *x*, *y* ∈ *S*, [*x*]_{(μ, t)}[*y*]_{(μ, t)} = [*xy*]_{(μ, t)}.

*Let* *S* = {0, *a*, *b*, *c*} *be a semigroup with the following* “*” *table*.

*Assume that*
$\mu =\frac{0.1}{0}+\frac{0.4}{a}+\frac{0.7}{b}+\frac{0.7}{c}$ *is a fuzzy ideal in S*. *Here* *t* =0.7, *then we have* *U*(*μ*;0.7)= {(0,0),(*a*, *a*),(*b*, *b*),(*c*, *c*),(*b*, *c*)}, *so we have* [0]_{(μ,0.7)}={0}, [*a*]_{(μ,0.7)}={*a*}, [*b*]_{(μ,0.7)}={*b*, *c*}, [*c*]_{(μ,0.7)}= {*b*, *c*}. *Obviously we can easily check* *U*(*μ*, *t*) *is a complete congruence relation*.

Let *μ* be a fuzzy ideal of *S* and *t* ∈ [0, 1]. We see that *U*(*μ*, *t*) is a congruence relation. According to Pawlak rough sets, we can obtain the approximation space (*S*, *μ*, *t*), just like (*U*, *θ*) in Pawlak rough sets.

*Let* *μ* *be a fuzzy ideal of* *S* *and* *U*(*μ*, *t*) *be a* *t*-*level set*. *For* *X* ⊆ *S* *and* *X* ≠ *∅*, *we define the upper and lower approximations over* (*S*, *μ*, *t*) *as follows*:
$$\underset{\_}{U}(\mu ,t,X)=\{x\in S\phantom{\rule{thinmathspace}{0ex}}|\phantom{\rule{thinmathspace}{0ex}}[x{]}_{(\mu ,t)}\subseteq X\}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{a}\mathit{n}\mathit{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\overline{U}(\mu ,t,X)=\{x\in S\phantom{\rule{thinmathspace}{0ex}}|\phantom{\rule{thinmathspace}{0ex}}[x{]}_{(\mu ,t)}\cap X\ne \mathrm{\varnothing}\}.$$

*If*
$\overline{U}(\mu ,t,X)(\underset{\_}{U}(\mu ,t,X))$ *is a subsemigroup of* *S*, *then we call* *X* *an upper* (*lower*) *rough subsemigroup of* *S*;

*If*
$\overline{U}(\mu ,t,X)(\underset{\_}{U}(\mu ,t,X))$
*is* *a* (*prime*) *ideal of* *S*, *then we call* *X* *an upper* (*lower*) *rough* (*prime*) *ideal of* *S*.

*Let* *μ* *be any fuzzy ideal of* *S*, *A and* *B* *be any nonempty subsets of* *S*, *then the following hold*:

$\underset{\_}{U}(\mu ,t,A)\subseteq A\subseteq \overline{U}(\mu ,t,A),$

$\overline{U}(\mu ,t,A\cap B)\subseteq \overline{U}(\mu ,t,A)\cap \overline{U}(\mu ,t,B),$

$\underset{\_}{U}(\mu ,t,A\cap B)=\underset{\_}{U}(\mu ,t,A)\cap \underset{\_}{U}(\mu ,t,B),$

$\overline{U}(\mu ,t,A\cup B)=\overline{U}(\mu ,t,A)\cup \overline{U}(\mu ,t,B),$

$\underset{\_}{U}(\mu ,t,A\cup B)\supseteq \underset{\_}{U}(\mu ,t,A)\cup \underset{\_}{U}(\mu ,t,B),$

$A\subseteq B,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{t}\mathit{h}\mathit{e}\mathit{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\overline{U}(\mu ,t,A)\subseteq \overline{U}(\mu ,t,B)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{a}\mathit{n}\mathit{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\underset{\_}{U}(\mu ,t,A)\subseteq \underset{\_}{U}(\mu ,t,B),$

$\mu \subseteq \nu ,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{t}\mathit{h}\mathit{e}\mathit{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\overline{U}(\mu ,t,A)\subseteq \overline{U}(\nu ,t,A)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{a}\mathit{n}\mathit{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\underset{\_}{U}(\mu ,t,A)\supseteq \underset{\_}{U}(\nu ,t,A),$

$\overline{U}(\mu ,t,\overline{U}(\mu ,t,A))=\overline{U}(\mu ,t,A),$

$\underset{\_}{U}(\mu ,t,\underset{\_}{U}(\mu ,t,A))=\underset{\_}{U}(\mu ,t,A),$

$\overline{U}(\mu ,t,\underset{\_}{U}(\mu ,t,A))=\overline{U}(\mu ,t,A),$

$\underset{\_}{U}(\mu ,t,\overline{U}(\mu ,t,A))=\underset{\_}{U}(\mu ,t,A).$

*Let* *μ* *be a fuzzy ideal of* *S* *and* *t* ∈ [0, 1]. *If* *A*_{1} *and* *A*_{2} *are nonempty subsets of* *S*, *then*
$$\overline{U}(\mu ,t,{A}_{1})\cdot \overline{U}(\mu ,t,{A}_{2})\subseteq \overline{U}(\mu ,t,{A}_{1}{A}_{2}).$$

Suppose that
$x\in \overline{U}(\mu ,t,{A}_{1})\cdot \overline{U}(\mu ,t,{A}_{2}),$ then there exists
${a}_{i}\in \overline{U}(\mu ,t,{A}_{i})(i=1,2),$ and *x* = *a*_{1}*a*_{2}, so there exists *x*_{i} ∈ *A*_{i}(*i*=1,2) such that *x*_{i} ∈ [*a*_{i}]_{(μ, t)} *A*_{j}(*i*=1,2). Since *U*(*μ*, *t*) is a congruence relation, we have *x*_{1}*x*_{2} ∈ [*a*_{1} *a*_{2}]_{(μ, t)}, *x*_{1}_{x}_{2} ∈ *A*_{1}*A*_{2} . So we have
${x}_{1}{x}_{2}\in [{a}_{1}{a}_{2}{]}_{(\mu ,t)}\cap {A}_{1}{A}_{2},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{so}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}x={x}_{1}{x}_{2}\in \overline{U}(\mu ,t,{A}_{1}{A}_{2}).$

Hence,
$\overline{U}(\mu ,t,{A}_{1})\cdot \overline{U}(\mu ,t,{A}_{2})\subseteq \overline{U}(\mu ,t,{A}_{1}{A}_{2}).$

Next, we give an example to prove that the converse of containment in Proposition 3.9 does not hold.

*Consider the subsets* {*a*} *and* {*c*} *of* *S* *as in Example 3.4. We have*
$\overline{U}(\mu ,0.4,a)\overline{U}(\mu ,0.4,c)=ac=b.\text{\hspace{0.17em}However}\phantom{\rule{thinmathspace}{0ex}}\overline{U}(\mu ,0.4,ac)=\{{b}_{.}d\}.$
*This shows that the converse of containment in Proposition 3.9 does not hold*.

*Let* U(*μ, t*) *be a complete congruence relation on S. If* *A*_{1}*A*_{2} *are non-empty subsets of* *S*, *then we have*
$\underset{\_}{U}(\mu ,{t}_{},{A}_{1})\cdot \underset{\_}{U}(\mu ,{t}_{},{A}_{2})\subseteq \underset{\_}{U}(\mu ,{t}_{},{A}_{1}{A}_{2}).$

Suppose that
$x\in \underset{\_}{U}({\mu}_{},{t}_{},{A}_{1})\cdot \underset{\_}{U}({\mu}_{},{t}_{},{A}_{2}),\phantom{\rule{thinmathspace}{0ex}}$
then there exists *a*_{i} ∈
${a}_{i}\in \underset{\_}{U}({\mu}_{},{t}_{},{A}_{i})(i={1}_{},2)$
such that *x* = *a*_{1}*a*_{2}, and because [*a*_{i}]_{(μ, t)} ⊆ *A*_{i}(*i* = 1,2). Since U(*μ, t*) is a complete congruence relation, we have [*a*_{1}]_{(μ, t)}[*a*_{2}]_{(μ, t)}=[*a*_{1}*a*_{2}]_{(μ,t)} ⊆ *A*_{1}*A*_{2}, and so *x* =*a*_{1}*a*_{2} ∈
$x={a}_{1}{a}_{2}\in \underset{\_}{U}(\mu ,{t}_{},{A}_{1}{A}_{2}).$

Hence,
$\underset{\_}{U}(\mu ,{t}_{},{A}_{1})\cdot \underset{\_}{U}(\mu ,{t}_{},{A}_{2})\subseteq \underset{\_}{U}(\mu ,{t}_{},{A}_{1}{A}_{2}).$

Here, we prove that if U(*μ,t*) is not a complete congruence relation on *S*, the containment in Proposition 3.11 may not be true as in the following example.

*Consider the subsets* {*b*, *c*} *and* {*b*, *c* *of* *S* *in Example 3.4. We have*
$\underset{\_}{U}(\mu ,{0.4}_{},\{{a}_{},c\})\underset{\_}{U}({\mu}_{},0.4,\{a,c\})=c\cdot c=b.\phantom{\rule{thinmathspace}{0ex}}\mathrm{However}\phantom{\rule{thinmathspace}{0ex}}\underset{\_}{U}({\mu}_{},0.4,\{b,c\}\{{b}_{},c\})=\underset{\_}{U}({\mu}_{},{0.4}_{},b)=\mathrm{\varnothing}.$
*This shows that the containment in Proposition 3.11 does not hold*.

In the following, we use the above conclusions to study the properties of rough subsemigroups and rough (prime) ideals.

*Let* *μ* *be a fuzzy ideal of* *S* *and* *t* ∈ [0, 1], *then*

*If A is a subsemigroup of* *S*, *A is an upper rough subsemigroup of* *S*;

*If A is an ideal of* *S*, *A is an upper rough ideal of* *S*.

Let *A* be a subsemigroup of *S*, we have *AA ⊆ A*, according to Propositions 3.8 and 3.9, we have
$\overline{U}(\mu ,{t}_{},A)\cdot \overline{U}(\mu ,{t}_{},A)\subseteq \overline{U}(\mu ,{t}_{},AA)\subseteq \overline{U}(\mu ,{t}_{},A),$
so we know that
$\overline{U}(\mu ,{t}_{},A)$
is a subsemigroup. By Definition 3.7, we have that *A* is an upper rough subsemigroup of *S*.

Let *A* be an ideal of *S*, then *SAS ⊆ A*. By Proposition 3.9, we have
$\overline{U}({\mu}_{},{t}_{},S)\cdot \overline{U}({\mu}_{},{t}_{},A)\cdot \overline{U}({\mu}_{},{t}_{},S)\subseteq \overline{U}({\mu}_{},{t}_{},SAS)\subseteq \overline{U}({\mu}_{},{t}_{},A),\text{so}\phantom{\rule{thinmathspace}{0ex}}\overline{U}({\mu}_{},{t}_{},A)$
is an ideal of *S*. Therefore, according to Definition 3.7, we have that *A* is an upper rough ideal of *S*.

*Consider the subset* {*c*, *d*} *as in Example 3.4. We have*
$\overline{U}(\mu ,0.4,\{c,d\})=\{{b}_{},{c}_{},d\},\phantom{\rule{thinmathspace}{0ex}}\mathrm{here}\phantom{\rule{thinmathspace}{0ex}}\{{b}_{},{c}_{},d\}$
*is a subsemigroup, so*
$\overline{U}(\mu ,{0.4}_{},\{{c}_{},d\})$
*is an upper rough subsemigroup, but* {*c*, *d*} *is not a subsemigroup*.

*Let* U(*μ, t*) *be a complete congruence relation on* *S*, *then*

*If A is a subsemigroup of* *S*, *A is a lower rough subsemigroup of* *S*;

*If A is an ideal of* *S*, *A is a lower rough ideal of* *S*.

It is similar to the proof of Theorem 3.13. The following example shows that even if *A* is not a subsemigroup of *S*, then
$\underset{\_}{U}({\mu}_{},{t}_{},A)$
may be a subsemigroup of *S* when U(*μ, t*) is complete congruence relation on *S*.

*Consider the semigroup* *S* *and the congruence class as in Example 3.6. Then* *A* = {0, *a*, *c*} *is not a subsemigroup of* *S* *but*
$\underset{\_}{U}(\mu ,{t}_{},A)=\{{0}_{},a\}$
*is a subsemigroup of* *S*.

*Let* *μ* *be a fuzzy ideal of* *S* *and* *t* ∈ [0, 1]. *Then* [0]_{(μ,t)} *is an ideal of* *S*.

For all *x*_{1}, *x*_{2} ∈ *S* and *a* ∈ [0]_{(μ, t)}, then we have *x*_{1}*ax*_{2} ∈ [*x*_{1}]_{(μ, t)}[0]_{(μ, t)}[*x*_{2}]_{(μ, t)} ⊆ [*x*_{1}}0*x*_{2}]_{(μ, t)}= [0]_{(μ, t)}. This means *x*_{1}*ax*_{2} ∈ [0]_{(μ, t)}. Hence [0]_{(μ, t)} is an ideal of *S*.

*Let μ be a fuzzy ideal of S and*
$t\in [{0}_{},\mu (0)].\mathit{T}\mathit{h}\mathit{e}\mathit{n}\phantom{\rule{1em}{0ex}}\underset{\_}{U}(\mu ,{t}_{},[0{]}_{({\mu}_{},t)})=[0{]}_{({\mu}_{},t)}.$

By Proposition 3.8, we have
$\underset{\_}{U}({\mu}_{},t,[0{]}_{({\mu}_{},t)})\subseteq [0{]}_{({\mu}_{},t)}.$
Now, we show that
$[0{]}_{(\mu ,t)}\subseteq \underset{\_}{U}(\mu ,t,[0{]}_{(\mu ,t)}).$

For every *x*∈[0]_{(μt)}, then(0, *x*) ∈*U(μ,t)*. Let*y* ∈[*x*]_{(μ, t)}, then *(x,y)* ∈ *U(μ,t)* . Since *U(μ,t*) is acongruence relation, we have (0, *y*)∈ *U(μ,t*), this implies *y* ∈*[0]*_{μ,t}) Hence
$[x{]}_{(\mu ,t)}\subseteq [0{]}_{(\mu ,t)}.$
This means
$x\in \underset{\_}{U}(\mu ,t,[0{]}_{(\mu ,t)}).$
Therefore
$\underset{\_}{U}(\mu ,t,[0{]}_{(\mu ,t)})=[0{]}_{(\mu ,t)}.$

*Let μ be a fuzzy ideal of S and*
$t\in [0{,}_{\mu (0)}].$
*then* [0]_{(μ,t)}*is a lower rough ideal of S*.

*Let **μ* be a fuzzy ideal of *S* and *t* ∈ [0, *μ*(0)].*Then μ*_{t}=[0](_{μ,t}).

For all *x* ∈*μ*_{t}, then *μ(x)*≥*t* and *μ*(0)≥*μ(x)*≥*t*. So (*μ(x*)∧*μ*(0))∨ Id_{S}(*x*,0)≥*μ(x)*∧*μ*(0)∧*t*, by Definition 3.1, we have (*x*,O)∈ *U(μ,t*). It implies *x* ∈[0]_{(μ,t}). This means
${\mu}_{t}\subseteq [0{]}_{({\mu}_{},t)}.$

On the other hand, for all *x* ∈ [0]_{μ,t}), then (*x*,O)∈ *U*(*μ*, *t*) and(*μ x*)∧*μ*(0))∨ *Id*_{S(x},0)≥*t*. If *x* = 0*μ*(*x*) = *μ*(0) ≥ *t* is obvious, so *x* ∈ *μ*_{t}, it implies [0]_{(μ,t)} ⊆ *μ*_{t}. If *x* ≠ 0, then *μ*(*x*)∧ *μ*(0)≥ *t*, *μ*(*x*) ≥ *t*, this means *x* ∈ *μ*_{t}. Hence [0]_{(μ, t)} ⊆ *μ*_{t}. Therefore *μ*_{t} = [0]_{(μ,t)}.

*Let U*(*μ*, *t*) *be a complete congruence relation on S. If A is a prime ideal of S, then*
$\underset{\_}{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A)$
*is a prime ideal of S if*
$\underset{\_}{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A)\ne \mathrm{\varnothing}.$

Since *A* is an ideal of *S*, by Proposition 3.16, we know that
$\underset{\_}{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A)$
is an ideal of *S*. Let
${x}_{1},{x}_{2}\in \underset{\_}{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A)$
for some *x*_{1}, *x*_{2} ∈ *S*, then we have [*x*_{1}]_{(μ,t)}[*x*_{2}]_{(μ,t)} ⊆ [*x*_{1}*x*_{2}]_{(μ,t)} ⊆ *A*.We suppose that
$\underset{\_}{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A)$
is not a prime ideal, then there exist *x*_{1}, *x*_{2} ∈ *S* such that
${x}_{1}{x}_{2}\in \underset{\_}{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{but}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{x}_{1}\notin \underset{\_}{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A),{x}_{2}\notin \underset{\_}{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A).$
Thus [*x*_{1}]_{(μ,t)} ⊈ *A*,[*x*_{2}]_{(μ,t)} ⊈ *A*, then exist
${x}_{1}^{\prime}\in [{x}_{1}{]}_{(\mu ,t)},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{x}_{1}^{\prime}\notin A,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{x}_{2}^{\prime}\in [{x}_{2}{]}_{(\mu ,t)},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{x}_{2}^{\prime}\notin A.\text{Thus}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{x}_{1}^{\prime}{x}_{2}^{\prime}\in $
[*x*_{1}]_{(μ, t)}[*x*_{2}]_{(μ, t)} *⊆ A*. Since *A* is a prime ideal of *S*, we have
${x}_{i}^{\prime}\in A$
for some 1 ≤ i ≤ 2. It contradicts with the supposition. Hence
$\underset{\_}{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A)\ne \mathrm{\varnothing}$
is a prime ideal of *S*.

*Let U*(*μ*, *t*) *be a complete congruence relation on S. If A is a prime ideal of S*, *then A is an upper rough prime ideal of S*.

Since *A* is aprime ideal of *S*, by Theorem 3.13, we know that
$\overline{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A)$
is an ideal of *S*. Let
${x}_{1}{x}_{2}\in \overline{U}(\mu ,\phantom{\rule{thinmathspace}{0ex}}t,\phantom{\rule{thinmathspace}{0ex}}A)$
for some *x*_{1}, *x*_{2} *∈ S*, then we have [*x*_{1}*x*_{2}]_{(μ, t)} ∩ *A* = [*x*_{1}]_{(μ, t)}[*x*_{2}]_{(μ, t)} ∩ *A* ≠ 0. So there exist
${x}_{1}^{\prime}\in [{x}_{1}{]}_{(\mu ,t)},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{x}_{2}^{\prime}\in [{x}_{2}{]}_{(\mu ,t)}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{such that}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{x}_{1}^{\prime}{x}_{2}^{\prime}\in A.$
Since *A* is a prime ideal, we have
${x}_{i}^{\prime}\in A$
for some 1 ≤ i ≤ 2. Thus
$[{x}_{i}{]}_{(\mu ,t)}\cap A\ne \mathrm{\varnothing},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and so}\phantom{\rule{thinmathspace}{0ex}}{x}_{i}\in \overline{U}(\mu ,t,A)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and so}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\overline{U}(\mu ,t,A)$
is aprime ideal of *S*. This means that *A* is an upper rough prime ideal of *S*.

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