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Volume 14, Issue 1 (Jan 2016)

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Rough semigroups and rough fuzzy semigroups based on fuzzy ideals

Qiumei Wang
  • Department of Mathematics, Hubei University for Nationalities, Enshi, 445000, China
  • Email:
/ Jianming Zhan
  • Corresponding author
  • Department of Mathematics, Hubei University for Nationalities, Enshi, 445000, China
  • Email:
Published Online: 2016-12-30 | DOI: https://doi.org/10.1515/math-2016-0102

Abstract

In this paper, we firstly introduce a special congruence relation U(μ, t) induced by a fuzzy ideal μ in a semigroup S. Then we define the lower and upper approximations based on a fuzzy ideal in semigroups. We can establish rough semigroups, rough ideals, rough prime ideals, rough fuzzy semigroups, rough fuzzy ideals and rough fuzzy prime ideals according to the definitions of rough sets and rough fuzzy sets. Furthermore, we shall consider the relationships among semigroups and rough semigroups, fuzzy semigroups and rough fuzzy semigroups, and some relative properties are also discussed.

Keywords: Fuzzy ideal; t-level set; Rough semigroup; Rough fuzzy semigroup; Rough (prime) ideal; Rough fuzzy (prime) ideal

MSC 2010: 20N20; 16Y99

1 Introduction

In 1965, the notion of fuzzy sets was introduced by Zadeh [1] and he gave the way to study uncertainty problems. Thereafter, the theory has developed quickly and has been applied in many areas of our real world. Up to now, fuzzy mathematics has remained an important branch of mathematics. Meanwhile, in the early 20th century, some researchers began to research semigroups formally. Since 1950, the investigations of finite semigroups have become rather important in theoretical computer science because there are natural relations between finite semigroups and finite automatons. With the fuzzy mathematics being deeply studied, fuzzy algebras were also investigated. The study of fuzzy algebras is also one of the most important directions in fuzzy mathematics. In particular, a great number of studies on fuzzy groups and fuzzy semigroups were performed. It should be pointed out that the studies on fuzzy semigroups are deeply influencing the development of fuzzy theory. In recent years, a large number of fuzzy algebras have been developed by some scholars, for instances see [24]. Especially, Davvaz [5], Dudek [6] and Zhan [7] applied fuzzy theory to n-ary semigroups. Recently, Zhan [8, 9] studied some types of fuzzy k-ideals and fuzzy h-ideals of hemirings, respectively.

In 1982, Pawlak [10] firstly developed a rough set theory. It is rather important to deal with inexact, uncertain, or vague situations. More and more researchers have been attracted to study rough sets and their applications. While mentioning rough sets, we can not neglect the lower and upper approximations. In Pawlak rough sets, the equivalent relations are essential to construct the lower and upper approximations. To deal with imperfect data, rough set theory is a powerful theory [11]. In recent years, Bonikowaski [12], Iwinski [13], and Pomykala [14] have studied algebraic properties of rough sets. The notion of rough subgroups was introduced by Biswas and Nanda [15]. Then, Kuroki [16] put up the notion of rough ideals in semigroups. Also, in [17], Jun investigated the roughness of Γ-subsemigroups and ideals in Γ-semigroups. The notions of rough prime ideals and rough fuzzy prime ideals in semigroups were introduced in [18].

It is obvious that congruence relations in rough sets are rather important. In Pawlak rough sets, congruence relations are necessary in the constructions of the lower and upper approximations. Therefore, if we try to investigate the rough sets, we should define a congruence relation. In recent years, some researchers try to find something to serve as a congruence relation in some algebraic structures. For examples, Kuroki and Wang [19] made use of a normal subgroup of groups as a congruence relation to discuss the properties of the lower and upper approximations. In [20], Davvaz proved that an ideal of rings could be regarded as a congruence relation. Further, Davvaz studied rough subrings and rough ideals in rings. Especially, in [21] Davvaz constructed the t-level relation based on a fuzzy ideal μ and he proved that U(μ; t) is a congruence relation in rings.

Based on the above idea, in this paper we will study the properties of rough subsemigroups and rough fuzzy subsemigroups of semigroups. We construct the t-level relation U(μ; t) of a fuzzy ideal μ of semigroups and prove that U(μ; t) is a congruence relation. We divide this paper into four parts. In section 2, we recall some basic results. In section 3, based on this congruence relation, we construct rough subsemigroup and rough (prime) ideals of semigroups. Some examples are also presented. Finally, we investigate rough fuzzy semigroups of semigroups based on this congruence relation in section 4.

2 Preliminaries

In this section we will recall some basic definitions that can be used in this paper.

Definition 2.1: A semigroup S is a nonempty set with a binary operation “*” such that

  1. a * bS, for all a, bS.

  2. (a * b)*c = a*(b * c), for all a, b, cS.

Definition 2.2: ([18]). Let S be a semigroup, a subset A of S is called a subsemigroup of S if A is closed under “*” and SAA(ASA); If A is not only a left ideal but also is a right ideal, it is a bi-ideal; A is called a prime ideal if A is ideal, and if a * bA, we can get aA or b ∈ A, for all a, bA.

Definition 2.3: ([18]). A fuzzy set μ of S is called a fuzzy subsemigroup of S if it satisfies μ(x * y) ≥ μ(x)∧ μ(y); μ is called a fuzzy let (right) ideal if it satisfies (1) μ(x * y) ≥ μ(x)∧ μ(y)(2)μ(x * y) ≥ μ(y)(μ(x * y) ≥ μ(x)), for all x, yS; μ is called a fuzzy ideal if it is not only a fuzzy let ideal but also is a fuzzy right ideal; μ is called a fuzzy prime ideal if μ(x * y) = μ(x)orμ(x * y) = μ(y) ,for all x,yS.

Definition 2.4: ([3]). Let R be an equivalence relation in a set A, if R is a binary relation and it satisfies

  1. Reflexive:for any aA,(a, a) ∈ R.

  2. Symmetry:for all a,bA,(a, b) ∈ R, then (b, a) ∈ R.

  3. Transitivity:for all a, b, cA,(a, b) ∈ R,(b, c) ∈ R, then (a, c) ∈ R.

An equivalence relation R is called a congruence relation if (a, b) ∈ R, then (ax, bx) ∈ R and (xa, xb) ∈ R, for all xA.

3 Rough semigroups based on fuzzy ideals

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.

Definition 3.1: 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 dS(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; xy, then I ds(x, y) = 0. Next, we prove U(μ, t)is a congruence relation.

Lemma 3.2: 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.

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

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

  3. 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 = yz, 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 xy = 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 xyz, 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 axay, then xy, then (μ(ax)μ(ay))Ids(ax,ay)=μ(ax)μ(ay)>_[μ(a)μ(x)][μ(a)μ(y)]=[μ(a)μ(a)][μ(x)μ(a)][μ(a)μ(y)][μ(x)μ(y)]>_μ(x)μ(y)>_tThus, 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 xt y(mod μ). If for elements x, yS, 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.

Lemma 3.3: Let μ be a fuzzy ideal of S and t ∈ [0, 1], then [x](μ, t)[y](μ, t) ⊆ [xy](μ, t).

Example 3.4: Let S = {a, b, c, d} be a semigroup with the following “*” table.

Assume that μ=0.3a+0.5b+0.1c+0.8d 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.

Definition 3.5: U(μ, t) is called a complete congruence relation if it satisfies: for any x, yS, [x](μ, t)[y](μ, t) = [xy](μ, t).

Example 3.6: Let S = {0, a, b, c} be a semigroup with the following “*” table.

Assume that μ=0.10+0.4a+0.7b+0.7c 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.

Definition 3.7: Let μ be a fuzzy ideal of S and U(μ, t) be a t-level set. For XS and X, we define the upper and lower approximations over (S, μ, t) as follows: U_(μ,t,X)={xS|[x](μ,t)X}andU¯(μ,t,X)={xS|[x](μ,t)X}.

  1. If U¯(μ,t,X)(U_(μ,t,X)) is a subsemigroup of S, then we call X an upper (lower) rough subsemigroup of S;

  2. If U¯(μ,t,X)(U_(μ,t,X)) is a (prime) ideal of S, then we call X an upper (lower) rough (prime) ideal of S.

Proposition 3.8: Let μ be any fuzzy ideal of S, A and B be any nonempty subsets of S, then the following hold:

  1. U_(μ,t,A)AU¯(μ,t,A),

  2. U¯(μ,t,AB)U¯(μ,t,A)U¯(μ,t,B),

  3. U_(μ,t,AB)=U_(μ,t,A)U_(μ,t,B),

  4. U¯(μ,t,AB)=U¯(μ,t,A)U¯(μ,t,B),

  5. U_(μ,t,AB)U_(μ,t,A)U_(μ,t,B),

  6. AB,thenU¯(μ,t,A)U¯(μ,t,B)andU_(μ,t,A)U_(μ,t,B),

  7. μν,thenU¯(μ,t,A)U¯(ν,t,A)andU_(μ,t,A)U_(ν,t,A),

  8. U¯(μ,t,U¯(μ,t,A))=U¯(μ,t,A),

  9. U_(μ,t,U_(μ,t,A))=U_(μ,t,A),

  10. U¯(μ,t,U_(μ,t,A))=U¯(μ,t,A),

  11. U_(μ,t,U¯(μ,t,A))=U_(μ,t,A).

Proof: The proofs are trivial.

Proposition 3.9: Let μ be a fuzzy ideal of S and t ∈ [0, 1]. If A1 and A2 are nonempty subsets of S, then U¯(μ,t,A1)U¯(μ,t,A2)U¯(μ,t,A1A2).

Proof: Suppose that xU¯(μ,t,A1)U¯(μ,t,A2), then there exists aiU¯(μ,t,Ai)(i=1,2), and x = a1a2, so there exists xiAi(i=1,2) such that xi ∈ [ai](μ, t) Aj(i=1,2). Since U(μ, t) is a congruence relation, we have x1x2 ∈ [a1 a2](μ, t), x1x2A1A2 . So we have x1x2[a1a2](μ,t)A1A2,sox=x1x2U¯(μ,t,A1A2).Hence, U¯(μ,t,A1)U¯(μ,t,A2)U¯(μ,t,A1A2).Next, we give an example to prove that the converse of containment in Proposition 3.9 does not hold.

Example 3.10: Consider the subsets {a} and {c} of S as in Example 3.4. We have U¯(μ,0.4,a)U¯(μ,0.4,c)=ac=b. HoweverU¯(μ,0.4,ac)={b.d}. This shows that the converse of containment in Proposition 3.9 does not hold.

Proposition 3.11: Let U(μ, t) be a complete congruence relation on S. If A1A2 are non-empty subsets of S, then we have U_(μ,t,A1)U_(μ,t,A2)U_(μ,t,A1A2).

Proof: Suppose that xU_(μ,t,A1)U_(μ,t,A2), then there exists aiaiU_(μ,t,Ai)(i=1,2) such that x = a1a2, and because [ai](μ, t)Ai(i = 1,2). Since U(μ, t) is a complete congruence relation, we have [a1](μ, t)[a2](μ, t)=[a1a2](μ,t)A1A2, and so x =a1a2x=a1a2U_(μ,t,A1A2).Hence, U_(μ,t,A1)U_(μ,t,A2)U_(μ,t,A1A2).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.

Example 3.12: Consider the subsets {b, c} and {b, c of S in Example 3.4. We have U_(μ,0.4,{a,c})U_(μ,0.4,{a,c})=cc=b.HoweverU_(μ,0.4,{b,c}{b,c})=U_(μ,0.4,b)=. 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.

Theorem 3.13: Let μ be a fuzzy ideal of S and t ∈ [0, 1], then

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

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

Proof:

  1. Let A be a subsemigroup of S, we have AA ⊆ A, according to Propositions 3.8 and 3.9, we have U¯(μ,t,A)U¯(μ,t,A)U¯(μ,t,AA)U¯(μ,t,A), so we know that U¯(μ,t,A) is a subsemigroup. By Definition 3.7, we have that A is an upper rough subsemigroup of S.

  2. Let A be an ideal of S, then SAS ⊆ A. By Proposition 3.9, we have U¯(μ,t,S)U¯(μ,t,A)U¯(μ,t,S)U¯(μ,t,SAS)U¯(μ,t,A),soU¯(μ,t,A) is an ideal of S. Therefore, according to Definition 3.7, we have that A is an upper rough ideal of S.

Remark 3.14: The above theorem shows that the notion of an upper rough subsemigroup (ideal) is an extended notion of a usual subsemigroup (ideal) of a semigroup. The following example shows that the converse of Theorem 3.13 does not hold in general.

Example 3.15: Consider the subset {c, d} as in Example 3.4. We have U¯(μ,0.4,{c,d})={b,c,d},here{b,c,d} is a subsemigroup, so U¯(μ,0.4,{c,d}) is an upper rough subsemigroup, but {c, d} is not a subsemigroup.

Proposition 3.16: Let U(μ, t) be a complete congruence relation on S, then

  1. If A is a subsemigroup of S, A is a lower rough subsemigroup of S;

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

Proof: 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 U_(μ,t,A) may be a subsemigroup of S when U(μ, t) is complete congruence relation on S.

Example 3.17: 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 U_(μ,t,A)={0,a} is a subsemigroup of S.

Lemma 3.18: Let μ be a fuzzy ideal of S and t ∈ [0, 1]. Then [0](μ,t) is an ideal of S.

Proof: For all x1, x2S and a ∈ [0](μ, t), then we have x1ax2 ∈ [x1](μ, t)[0](μ, t)[x2](μ, t) ⊆ [x1}0x2](μ, t)= [0](μ, t). This means x1ax2 ∈ [0](μ, t). Hence [0](μ, t) is an ideal of S.

Proposition 3.19: Let μ be a fuzzy ideal of S and t[0,μ(0)].ThenU_(μ,t,[0](μ,t))=[0](μ,t).

By Proposition 3.8, we have U_(μ,t,[0](μ,t))[0](μ,t). Now, we show that [0](μ,t)U_(μ,t,[0](μ,t)).For every x∈[0](μt), then(0, x) ∈U(μ,t). Lety ∈[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](μ,t)[0](μ,t). This means xU_(μ,t,[0](μ,t)). Therefore U_(μ,t,[0](μ,t))=[0](μ,t).

Corollary 3.20: Let μ be a fuzzy ideal of S and t[0,μ(0)]. then [0](μ,t)is a lower rough ideal of S.

Proposition 3.21: Let μ be a fuzzy ideal of S and t ∈ [0, μ(0)].Then μt=[0](μ,t).

For all xμt, then μ(x)t and μ(0)≥μ(x)t. So (μ(x)∧μ(0))∨ IdS(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 μt[0](μ,t).On the other hand, for all x ∈ [0]μ,t), then (x,O)∈ U(μ, t) and(μ x)∧μ(0))∨ IdS(x,0)≥t. If x = 0μ(x) = μ(0) ≥ t is obvious, so xμt, it implies [0](μ,t)μt. If x ≠ 0, then μ(x)∧ μ(0)≥ t, μ(x) ≥ t, this means xμt. Hence [0](μ, t)μt. Therefore μt = [0](μ,t).

Theorem 3.22: Let U(μ, t) be a complete congruence relation on S. If A is a prime ideal of S, then U_(μ,t,A) is a prime ideal of S if U_(μ,t,A).

Since A is an ideal of S, by Proposition 3.16, we know that U_(μ,t,A) is an ideal of S. Let x1,x2U_(μ,t,A) for some x1, x2S, then we have [x1](μ,t)[x2](μ,t) ⊆ [x1x2](μ,t)A.We suppose that U_(μ,t,A) is not a prime ideal, then there exist x1, x2S such that x1x2U_(μ,t,A)butx1U_(μ,t,A),x2U_(μ,t,A). Thus [x1](μ,t)A,[x2](μ,t)A, then exist x1[x1](μ,t),x1A,x2[x2](μ,t),x2A.Thusx1x2 [x1](μ, t)[x2](μ, t) ⊆ A. Since A is a prime ideal of S, we have xiA for some 1 ≤ i ≤ 2. It contradicts with the supposition. Hence U_(μ,t,A) is a prime ideal of S.

Theorem 3.23: 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 U¯(μ,t,A) is an ideal of S. Let x1x2U¯(μ,t,A) for some x1, x2 ∈ S, then we have [x1x2](μ, t)A = [x1](μ, t)[x2](μ, t)A ≠ 0. So there exist x1[x1](μ,t),x2[x2](μ,t)such thatx1x2A. Since A is a prime ideal, we have xiA for some 1 ≤ i ≤ 2. Thus [xi](μ,t)A,and soxiU¯(μ,t,A)and soU¯(μ,t,A) is aprime ideal of S. This means that A is an upper rough prime ideal of S.

4 Rough fuzzy semigroups based on fuzzy ideals

In Section 3, we discuss rough semigroups and their properties of semigroups. Now, in this section, we introduce roughness of fuzzy semigroups based on a fuzzy ideal of semigroups. First of all, we give the definition of rough fuzzy semigroups of semigroups. Here, we also use U(μ, t) as a congruence relation.

Definition 4.1: Let U(μ, t) be a complete congruence relation on S and β be a fuzzy subset of S, we can define a new approximation space (μ,t,β) and the lower and upper approximations of β, respectively U¯(μ,t,β)={β(y)|y[x](μ,t,)}andU_(μ,t,β)={β(y)|y[x](μ,t)}.

  1. ifU¯(μ,t,β)(U_(μ,t,β)) is a fuzzy subsemigroup of S, then we call β an upper (lower) rough fuzzy subsemigroup of S;

  2. ifU¯(μ,t,β)(U_(μ,t,β)) is a fuzzy left (right, two-sided) ideal of S, then we call β an upper (lower) rough fuzzy left (right, two-sided) ideal of S.

Proposition 4.2: Let μ be any fuzzy ideal of S and β and γ be any two fuzzy subsets, then

  1. U_(μ,t,β)βU¯(μ,t,β),

  2. U¯(μ,t,βγ)U¯(μ,t,β)U¯(μ,t,γ),

  3. U_(μ,t,βγ)=U_(μ,t,β)U_(μ,t,γ),

  4. U¯(μ,t,βγ)=U¯(μ,t,β)U¯(μ,t,γ),

  5. U_(μ,t,βγ)U_(μ,t,β)U_(μ,t,γ),

  6. βγ,thenU¯(μ,t,β)U¯(μ,t,γ)andU_(μ,t,β)U_(μ,t,γ),

  7. U¯(μ,t,U¯(μ,t,β))=U¯(μ,t,β),

  8. U_(μ,t,U_(μ,t,β))=U_(μ,t,β),

  9. U¯(μ,t,U_(μ,t,β))=U¯(μ,t,β),

  10. U_(μ,t,U¯(μ,t,β))=U_(μ,t,β),

  11. U¯(μ,t,βα)=U¯(μ,t,β)α,

  12. U_(μ,t,βα)=U_(μ,t,β)α,

  13. U¯(μ,t,βα¯)=U¯(μ,t,β)α¯,

  14. U_(μ,t,βα¯)=U_(μ,t,β)α¯.

Here we only prove that (12) and (13) hold and others are trivial. xU_(μ,t,β)αU_(μ,t,β)(x)>_α{β(y)|y[x](μ,t)}>_αy[x](μ,t),thenβ(y)>_αy[x](μ,t),Yβα[x](μ,t)βαxU_(μ,t,βα).xU¯(μ,t,β)αU¯(μ,t,β)(x)>_α{β(y)|y[x](μ,t)}>_αthere exists y ∈[x](μ,t), and β(y)≥ α and there exists y ∈ [x](μ,t), and yβαxU¯(μ,t,βα¯).

Theorem 4.3: Let μ be a fuzzy ideal of S and t[0, 1], then

  1. If β is a fuzzy subsemigroup of S, β is an upper rough fuzzy subsemigroup of S;

  2. If β is a fuzzy left (right, two-sided) ideal of S, β is an upper rough fuzzy left ( right, two-sided) ideal of S.

  1. Let β be a fuzzy subsemigroup of S, for every α [0, 1], βα is a subsemigroup of S. According to Propositions 3.9 and 4.2, we have U¯(μ,t,β)αU¯(μ,t,β)α=U¯(μ,t,βα)U¯(μ,t,βα)U¯(μ,t,βαβα)U¯(μ,t,βα)=U¯(μ,t,β)α, obviously, for any α ∈ [0, 1], U¯(μ,t,β)α is a subsemigroup of S. So U¯(μ,t,β) is a fuzzy subsemigroup of S. Thus, β is an upper rough fuzzy subsemigroup of S.

  2. If β is a fuzzy left ideal of S, for every α [0, 1], βα is a left ideal of S, that is, S · βαβα, since U¯μ,t,S=S,SU¯μ,t,βα=U¯μ,t,SU¯μ,t,βαU¯μ,t,SβαU¯μ,t,βα=U¯μ,t,Aα, so U¯(μ,t,β)α is aleft ideal of S, then U¯(μ,t,β) is a fuzzy left ideal of S. Hence β is an upper rough fuzzy left ideal of S.

From this theorem, we see that upper rough fuzzy subsemigroups (left ideals, right ideals, two-sided ideals) are the generalizations of the fuzzy subsemigroups (left ideals, right ideals, two-sided ideals).

Proposition 4.4: Let U(μ,t) be a complete congruence relation on S, then

  1. If β is a fuzzy subsemigroup of S, β is a lower rough fuzzy subsemigroup of S;

  2. If β is a fuzzy left (right, two-sided) ideal of S, β is a lower rough fuzzy left ( right, two-sided) ideal of S.

  1. Let β be a fuzzy subsemigroup of S, for every α ∈ [0, 1], βα is a subsemigroup of S. According to Propositions 3.9 and 4.2, we have U_μ,t,βαU_μ,t,βα = U_μ,t,βαU_μ,t,βαμ,t,βαβαU_μ,t,βα=U_μ,t,βα, obviously, for any α ∈ [0, 1], U¯μ,t,βα is a subsemigroup of S. So U¯μ,t,β is a fuzzy subsemigroup of S. So β is alower rough fuzzy subsemigroup of S.

  2. If β is a fuzzy left ideal of S, for every α ∈ [0, 1], βα is a left ideal of S, that is S βα ⊆ βα, since U_μ,t,S=S,SU_μ,t,Aα=U_μ,t,SU_μ,t,βαU_μ,t,SβαU_μ,t,βα=U_μ,t,βα, so U_μ,t,βα is a left ideal of s, then U_μ,t,β is afuzzy left ideal of S. Hence β is an upper rough fuzzy left ideal.

Theorem 4.5: Let U(μt) be a complete congruence relation on S. If β is a fuzzy prime ideal of S, then we have

  1. If U_μ,t,β,β is an lower rough fuzzy prime ideal of S.

  2. If U¯μ,t,β,β is an upper rough fuzzy prime ideal of S.

  1. Since β is a fuzzy prime ideal of S, we know βα(α ∈ [0,1]) is a prime ideal of S. By Theorem 3.22 we have U_μ,t,βα, if it is nonempty, is a prime ideal of S. And by Proposition 4.2, we know U_μ,t,βα=U_μ,t,βα is also a prime ideal of S, if it is nonempty. So U_μ,t,β is a fuzzy prime ideal of S, if it is nonempty, then β is a lower rough fuzzy prime ideal of S.

  2. It can be proven as (1).

Acknowledgement

The authors are extremely grateful to the editor and the referees for their valuable comments and helpful suggestions which help to improve the presentation of this paper.

This research was supported by NNSFC (11561023).

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About the article

Received: 2016-09-02

Accepted: 2016-11-23

Published Online: 2016-12-30

Published in Print: 2016-01-01



Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0102. Export Citation

© 2016 Wang and Zhan. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

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