Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 2. pp. 43-52, 2011 Applied Mathematics
Fuzzifying Bi-Ideals of Semigroups and Implication Operators H. Hedayati∗, Z. Jafari
∗Department of Mathematics, Faculty of Basic Science, Babol University of
Technol-ogy, Babol, Iran
e-mail: hedayati143@ gm ail.com,hedayati143@ yaho o.com
Received Date: August 31, 2010 Accepted Date: January 25, 2011
Abstract. In this paper the notion of (α, β)−fuzzy bi-ideal of a semigroup is introduced and related properties are investigated. Also, the definition of implication operator in the Lukasiewicz system of continuous-valued logic for fuzzy ideals is considered. In particular, the relationship between fuzzy bi-ideals with thresholds and implication-based fuzzy bi-bi-ideals is analyzed. Key words: (α, β)-fuzzy bi-ideal; fuzzifying bi-ideal; implication-based fuzzy bi-ideal.
2000 Mathematics Subject Classification: 08A72. 1. Introduction
It is well known that semigroups appear in the classical relevant logics, some non-classical logics, and multi-modal arrow logics. The motivation to study finite semigroups appeared in the 1950s as a result of work on linguistics and models of computation and reasoning. From such works emerged the notion of a finite automaton of which several variants can be found in the literature. A rational subset of a semigroup S is a member of the smallest set of subsets of S which contains the empty set and the singleton subsets, and is closed under union, subset product, and taking the generated subsemigroup. By observing that the empty language and one letter languages are obviously recognizable, the above shows that every rational language over a finite alphabet is recognizable. Since the inception of the notion of a fuzzy set in 1965 [30] which laid the foundations of fuzzy set theory, the literature on fuzzy set theory and its applications has been growing rapidly amounting by now to several papers (see [1-29]). These are widely scattered over many disciplines such as artificial intelligence, computer science, control engineering, expert systems, management science, operations research, pattern recognition, robotics, and others.
After the introduction of fuzzy sets by Zadeh [30], reconsideration of some con-cepts of classical mathematics began. On the other hand, because of the impor-tance of group theory in mathematics, as well as its many areas of applications, the notion of a fuzzy subgroup was defined and its structure was investigated by Rosenfeld. This subject has been studied further by many mathematicians. The concept of a fuzzy ideal in semigroups was developed by Kuroki ([17-21]). He studied fuzzy ideals, fuzzy bi-ideals and fuzzy semiprime ideals in semi-groups. Fuzzy ideals and Green’s relations in semigroups were investigated by McLean and Kummer [23]. Dib and Galhum [9] introduced definitions of a fuzzy groupoid and a fuzzy semigroup, studied fuzzy ideals and fuzzy bi-ideals of a fuzzy semigroup. Fuzzy ideals, generated by fuzzy sets in semigroups, are considered by Mo and Wang [25] and Xiang [27]. A new type of fuzzy subgroups ((∈, ∈ ∨q)−fuzzy subgroups) was introduced in an earlier paper of Bhakat and Das [4] by using the combined notions of “belongingness” and “quasicoinci-dence” of fuzzy points and fuzzy sets. In fact, (∈, ∈ ∨q)−fuzzy subgroup is an important and useful generalization of Rosenfeld’s fuzzy subgroup. This concept has been studied further in [1], [2] and [5] . In [15] ([16]), Jun and Song (Kazanci, Yamak) discussed some fundamental aspects of (∈, ∈ ∨q)−fuzzy interior ideals (bi-ideals). They showed that ((∈, ∈ ∨q)−fuzzy interior ideals (bi-ideals) are generalization of the existing concepts fuzzy interior ideals (bi-ideals).
The aim of this paper is the introduce and study new sorts of fuzzy bi-ideals of a semigroup and investigate the new aspects of related properties. The combined notions of “belongingness” and “quasicoincidence” (in different cases) of fuzzy points and fuzzy sets were used to introduce these sorts of fuzzy bi-ideals. Also, the definition of implication operator in the Lukasiewicz system of continuous-valued logic for fuzzy bi-ideals was considered. In particular, the relationship between fuzzy bi-ideals with thresholds and implication-based fuzzy bi-ideals (under some important implication operators) is investigated.
2. Preliminaries
Let S be a semigroup. By a subsemigroup of S we mean a nonempty subset A of S such that A2⊆ A, and by a left(right) ideal of S such that SA ⊆ A(AS ⊆ A). By two sided ideal, we mean a nonempty subset of S which is both a left and right ideal of S. A subsemigroup A of a semigroup S is called a bi-ideal of S if ASA ⊆ A. A semigroup S is said to be right (resp. left) zero if xy = y (resp. xy = x) for all x, y ∈ S (see [17]).
Definition 2.1. [17] Let μ be a fuzzy subset of S and a, x, y, z ∈ S. (i) μ is called a fuzzy subsemigroup of S if μ(xy) ≥ μ(x) ∧ μ(y).
(ii) A fuzzy subsemigroup μ is called a fuzzy ideal of S if μ(xy) ≥ μ(x) ∨ μ(y).
(iii) A fuzzy subsemigroup μ is called a fuzzy bi-ideal of S if μ(xay) ≥ μ(x) ∧ μ(y).
Let μ be a fuzzy subset of a nonempty set S and t ∈ (0, 1]. The set U(μ; t) = {x ∈ S| μ(x) ≥ t} is called a level subset of μ.
Theorem 2.2. [17] Let S be a semigroup and let μ be a fuzzy bi-ideal of S. Then the level subset μt(6= ∅) is a bi-ideal of S for all t ∈ (0, 1] if and only if μ is a fuzzy bi-ideal of S.
Definition 2.3. [24] A fuzzy subset μ in a set S of the form μ(y) =
½
t 6= 0 if y = x, 0 if y 6= x
is said to be a fuzzy point with support x and value t and is denoted by xt. A fuzzy point xt is said to be belong to (resp. quasi-coincident with) a fuzzy set μ, written as xt∈ μ (resp. xtqμ) if μ(x) ≥ t (resp. μ(x) + t > 1). If xt∈ μ or xtqμ, then we write xt∈ ∨q μ. The symbol ∈ ∨q means neither ∈ nor q hold. Definition 2.4. [16] Let 0 ≤ s < t ≤ 1. Then a fuzzy set μ of S is called a fuzzy bi-ideal with thresholds (s, t) of S if for all a, x, y ∈ S, the following conditions hold:
(i) μ(xy) ∨ s ≥ μ(x) ∧ μ(y) ∧ t, (ii) μ(xay) ∨ s ≥ μ(x) ∧ μ(y) ∧ t.
Theorem 2.5. [16] A fuzzy set μ of S is a fuzzy bi-ideal with thresholds (s, t) of S if and only if μα(6= ∅) is a bi-ideal of S for all s < α ≤ t.
3. (α, β)-Fuzzy Bi-Ideals of Semigroups
Definition 3.1. A fuzzy subset μ of a semigroup S is called an (α, β)-fuzzy bi-ideal of S, where α 6=∈ ∧q, if for all t, r ∈ (0, 1] and x, y ∈ S,
(i) xt, yrαμ implies (xy)t∧rβμ,
(ii) xt, yrαμ and a ∈ S implies (xay)t∧rβμ.
Let μ be a fuzzy set in S such that μ(x) ≤ 0.5 for all x ∈ S. Let x ∈ S and t ∈ (0, 1] be such that xt∈ ∧qμ. Then μ(x) ≥ t and μ(x) + t > 1. It follows that
1 < μ(x) + t ≤ μ(x) + μ(x) = 2μ(x)
so that μ(x) > 0.5. This means that {xt|xt ∈ ∧qμ} = ∅. Therefore the case α =∈ ∧q in Definition 3.1 will be omitted.
Theorem 3.2. [16] Let μ be a fuzzy subset of a semigroup S. Then μ is an (∈, ∈ ∨q)-fuzzy bi-ideal of S if and only if the following conditions are hold for all a, x, y ∈ S:
(i) μ(xy) ≥ μ(x) ∧ μ(y) ∧ 0.5, (ii) μ(xay) ≥ μ(x) ∧ μ(y) ∧ 0.5.
Theorem 3.3. Let μ be a nonzero (α, β)-fuzzy bi-ideal of S. Then the set Supp(μ) = {x ∈ S|μ(x) > 0} is an bi-ideal of S.
Proof. Let x, y ∈ Supp(μ). Then μ(x) > 0 and μ(y) > 0. Assume that μ(xy) = 0. If α ∈ {∈, ∈ ∨q}, then xμ(x)αμ and yμ(y)αμ but (xy)μ(x)∧μ(y)βμ for every¯ β ∈ {∈, q, ∈ ∨q, ∈ ∧q}, a contradiction. Note that x1qμ and y1qμ but (xy)1∧1=
(xy)1βμ for every β ∈ {∈, q, ∈ ∨q, ∈ ∧q}, a contradiction. Hence μ(xy) > 0,¯ that is, xy ∈ Supp(μ). Now let x, y ∈ Supp(μ). Suppose that μ(xay) = 0 and let α ∈ {∈, ∈ ∨q}. Then xμ(x)αμ and yμ(y)αμ but (xay)μ(x)∧μ(y)βμ for¯ every β ∈ {∈, q, ∈ ∨q, ∈ ∧q}. This is a contradiction. Note that x1qμ and y1qμ but (xay)1∧1 = (xay)1βμ for every β ∈ {∈, q, ∈ ∨q, ∈ ∧q}, a contradiction.¯ Therefore μ(xay) > 0, and so xay ∈ Supp(μ). Consequently, Supp(μ) is an bi-ideal of S. ¤
Theorem 3.4. Let S be a right (resp. left) zero semigroup and let μ be a nonzero (q, q)-fuzzy bi-ideal of S. Then μ is constant on Supp(μ).
Proof. Let m be an element of S such that μ(m) = ∨{μ(x)|x ∈ S}. Then m ∈ Supp(μ). Suppose that there exists x ∈ Supp(μ) such that tx = μ(x) 6= μ(m) = tm. Then tx< tm. Choose r, s ∈ (0, 1] such that 1−tm< r < 1 −tx< s. Then mrqμ and xsqμ but (mx)r∧s= xrqμ(resp. (xm)¯ r∧s= xrqμ) because S is¯ right(resp. left) zero, a contradiction. Hence μ(x) = μ(e) for all x ∈ Supp(μ), and therefore μ is constant on Supp(μ).¤
Theorem 3.5. Let B be a bi-ideal of S and let μ be a fuzzy set in S such that (i) ∀x ∈ S \ B, μ(x) = 0,
(ii) ∀x ∈ B, μ(x) ≥ 0.5.
Then μ is a (q, ∈ ∨q)-fuzzy bi-ideal of S.
Proof. Let x, y ∈ S and t, r ∈ (0, 1] be such that xtqμ and yrqμ. Then x, y ∈ B, and so xy ∈ B. Thus if t ∧ r ≤ 0.5, then μ(xy) ≥ 0.5 ≥ t ∧ r and hence (xy)t∧r∈ μ. Therefore (xy)t∧r∈ ∨qμ. If t ∧ r > 0.5, then
μ(xy) + t ∧ r > 0.5 + 0.5 = 1
and so (xy)t∧rqμ. Therefore (xy)t∧r∈ ∨qμ. Now let x, a, y ∈ S and t, r ∈ (0, 1] be such that xtqμ and yrqμ. Then μ(x) + t > 1 and μ(y) + r > 1, which implies x, y ∈ B. Since B is a bi-ideal, it follows that xay ∈ B so that μ(xay) ≥ 0.5. If t ∧ r ≤ 0.5, then μ(xay) ≥ 0.5 ≥ t ∧ r and thus (xay)t∧r ∈ μ. Therefore (xay)t∧r∈ ∨qμ. If t ∧ r > 0.5, then
μ(xay) + t ∧ r > 0.5 + 0.5 = 1,
and so (xay)t∧rqμ. Hence (xay)t∧r∈ ∨qμ, and μ is a (q, ∈ ∨q)-fuzzy bi-ideal of S.¤
Theorem 3.6. Let S be a right or left zero semigroup and let μ be a (q, ∈ ∨q)-fuzzy bi-ideal of S such that μ is not constant on Supp(μ). If e is an element of S such that μ(e) = ∨{μ(x)| x ∈ S}, then μ(x) ≥ 0.5 for all x ∈ Supp(μ). Proof. Assume that μ(x) < 0.5 for all x ∈ S. Since μ is not constant on Supp(μ), there exists x ∈ Supp(μ) such that tx = μ(x) 6= μ(e) = te. Then tx < te. Choose t1 > 0.5 such that tx+ t1 < 1 < te+ t1. Then et1qμ and
x1qμ. Since μ(x) + t1= tx+ t1 < 1, we get xt1qμ and so (ex)¯ 1∧t1 = xt1∈ ∨qμ.
This contradictions the fact that μ is a (q, ∈ ∨q)-fuzzy bi-ideal of S. Therefore μ(x) ≥ 0.5 for some x ∈ S. Now let te = μ(e) < 0.5. Then there exists x ∈ S
such that tx = μ(x) ≥ 0.5. Thus te < tx, which is a contradiction. Hence μ(e) ≥ 0.5. Finally let tx = μ(x) < 0.5 for some x ∈ Supp(μ). Taking t1 > 0 such that tx+ t1< 0.5, then x1qμ and e0.5+t1qμ since μ(e) ≥ 0.5. But
μ(x) + 0.5 + t1= tx+ 0.5 + t1< 0.5 + 0.5 = 1,
which implies that x0.5+t1qμ. Thus (ex)¯ 1∧(0.5+t1)= x0.5+t1∈ ∨qμ, a
contradic-tion. Therefore μ(x) ≥ 0.5 for all x ∈ Supp(μ). ¤
A fuzzy set μ in S is said to be proper if Im(μ) has at least two elements. Two fuzzy sets are said to be equivalent if they have same family of level subsets. Otherwise, they are said to be non-equivalent.
Theorem 3.7. Let S have proper bi-ideals. A proper (∈, ∈)-fuzzy bi-ideal μ of S such that |Im(μ)| ≥ 3 can be expressed as the union of two proper non-equivalent (∈, ∈)-fuzzy bi-ideals of S.
Proof. Let μ be a proper (∈, ∈)-fuzzy bi-ideal of S with Im(μ) = {t0, t1, . . . , tn}, where t0> t1> . . . > tn and n ≥ 2. Then
U (μ; t0) ⊆ U(μ; t1) ⊆ . . . ⊆ U(μ; tn) = S
is the chain of ∈-level bi-ideals of S. Define fuzzy sets ν and θ in S by
ν(x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ r1 if x ∈ U (μ; t1) , t2 if x ∈ U (μ; t2) \U (μ; t1) , .. . tn if x ∈ U (μ; tn) \U (μ; tn−1) and θ(x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t0 if x ∈ U (μ; t0) , t1 if x ∈ U (μ; t1) \U (μ; t0) , r2 if x ∈ U (μ; t3) \U (μ; t1) , t4 if x ∈ U (μ; t4) \U (μ; t3) , .. . tn if x ∈ U (μ; tn) \U (μ; tn−1)
respectively, where t2< r1< t1and t4< r2< t2. Then ν and θ are (∈, ∈)-fuzzy bi-ideals of S with
U (μ; t1) ⊆ U(μ; t2) ⊆ . . . ⊆ U(μ; tn) = S and
U (μ; t0) ⊆ U(μ; t1) ⊆ U(μ; t3) . . . ⊆ U(μ; tn) = S
as respective chains of ∈-level bi-ideals, and ν, θ ≤ μ. Thus ν and θ are non-equivalent, and obviously ν ∪ θ = μ. This completes the proof. ¤
4. Fuzzy Bi-Ideals of Type (∈, ∈ ∨q)
Theorem 4.1. Every (∈ ∨q, ∈ ∨q)-fuzzy bi-ideal is an (∈, ∈ ∨q)-fuzzy bi-ideal. Proof.Let μ be an (∈ ∨q, ∈ ∨q)-fuzzy bi-ideal of S. Let x, y ∈ S and t, r ∈ (0, 1] be such that xt ∈ μ and yr ∈ μ. Then xt ∈ ∨qμ and yr ∈ ∨qμ. Since μ is an (∈ ∨q, ∈ ∨q)-fuzzy bi-ideal, it follows that (xy)t∧r ∈ ∨qμ. Now let x, a, y ∈ S and t, r ∈ (0, 1] be such that xt∈ μ and yr∈ μ. Then xt∈ ∨qμ and yr∈ ∨qμ, and so (xay)t∧r∈ ∨qμ. Consequently, μ is an (∈, ∈ ∨q)-fuzzy bi-ideal of S. ¤ Theorem 4.2. Every (∈, ∈)-fuzzy bi-ideal is an (∈, ∈ ∨q)-fuzzy bi-ideal. Proof. Straightforward. ¤
Example 4.3. We know that (N, +) is a semigroup. Define a fuzzy set μ in N as follows:
μ(x) = ½
0.7, if x ∈ 2N, 0.6, if x ∈ 2N
Then μ is an (∈, ∈ ∨q)-fuzzy bi-ideal of S, but μ is not an (∈, ∈)-fuzzy bi-ideal of S. Because if x, y ∈ 2N and a 6∈ 2N, then x0.7, y0.7∈ μ, but (x + a + y)0.7∈μ. Lemma 4.4. If B is a bi-ideal of S, then the characteristic function χB of B is an (∈, ∈)-fuzzy bi-ideal of S.
Proof. Let x, y ∈ S and t, r ∈ (0, 1] be such that xt∈ χB and yr∈ χB. Then χB(x) ≥ t > 0 and χB(y) ≥ r > 0, which imply that χB(x) = 1 = χB(y). Thus x, y ∈ B and so xy ∈ B. It follows that χB(xy) = 1 ≥ t∧r so that (xy)t∧r∈ χB. Now let x, a, y ∈ S and t, r ∈ (0, 1] be such that xt ∈ χB and yr ∈ χB. Then χB(x) ≥ t > 0 and χB(y) ≥ r > 0, and so χB(x) = 1 = χB(y), i.e., x, y ∈ B. Since B is a bi-ideal, we have xay ∈ B and hence χB(xay) = 1 ≥ t∧r. Therefore (xay)t∧r∈ χB and consequently χB is an (∈, ∈)-fuzzy bi-ideal of S. ¤
Theorem 4.5. For any subset B of S, χB is an (∈, ∈ ∨q)-fuzzy bi-ideal of S if and only if B is a bi-ideal of S.
Proof. Let χBbe an (∈, ∈ ∨q)-fuzzy bi-ideal of S. Let x, y ∈ B. Then x1∈ χB and y1∈ χB which imply that (xy)1= (xy)1∧1∈ ∨qχB. Hence χB(xy) > 0, and so xy ∈ B. Now let x, y ∈ B. Then χB(x) = 1 = χB(y), and thus x1, y1 ∈ χB. Since χB is an (∈, ∈ ∨q)-fuzzy bi-ideal, it follows that (xay)1 ∈ χB so that χB(xay) = 1. Hence xay ∈ B, and B is a ideal of S. Conversely, if B is a bi-ideal of S, then χB is an (∈, ∈)-fuzzy bi-ideal of S by Lemma 4.4, and therefore χB is an (∈, ∈ ∨q)-fuzzy bi-ideal of S by Theorem 4.2. ¤
Theorem 4.6. Let μ be an (∈, ∈ ∨q)-fuzzy bi-ideal of S such that μ(x) < 0.5 for all x ∈ S. Then μ is an (∈, ∈)-fuzzy bi-ideal of S.
Proof. Let x, y ∈ S and t, r ∈ (0, 1] be such that xt ∈ μ and yr ∈ μ. Then μ(x) ≥ t and μ(y) ≥ r. It follows from Theorem 3.2 that
so that (xy)t∧r∈ μ. Now let x, a, y ∈ S and t, r ∈ (0, 1] be such that xt, yr∈ μ. Then μ(x) ≥ t and μ(y) ≥ r and so
μ(xay) ≥ μ(x) ∧ μ(y) ∧ 0.5 = μ(x) ∧ μ(y) ≥ t ∧ r. Hence (xay)t∧r∈ μ and therefore μ is an (∈, ∈)-fuzzy bi-ideal of S. ¤ For any fuzzy set μ in S and t ∈ (0, 1], we denote
μt= {x ∈ S|xtqμ} and
[μ]t= {x ∈ S|xt∈ ∨qμ} Obviously, [μ]t= U (μ; t) ∪ μt.
Theorem 4.7. A fuzzy set μ in S is an (∈, ∈ ∨q)-fuzzy bi-ideal of S if and only if [μ]t is a bi-ideal of S for all t ∈ (0, 0.5]. We call [μ]t an (∈ ∨q)-level bi-ideal of μ.
Proof. Let μ be an (∈, ∈ ∨q)−fuzzy bi-ideal of S and x, y ∈ [μ]tfor t ∈ (0, 0.5]. Then xt ∈ ∨qμ and yt ∈ ∨qμ, which means μ(x) ≥ t or μ(x) + t > 1, and μ(y) ≥ t or μ(y) + t > 1. On the other hand by Theorem 3.2, we know μ(xy) ≥ μ(x) ∧ μ(y) ∧ 0.5, so we have μ(xy) ≥ t ∧ 0.5. Since if μ(xy) < t ∧ 0.5, then μ(x) ∧ μ(y) ∧ 0.5 ≤ μ(xy) < t ∧ 0.5, which implies μ(x) ∧ μ(y) ∧ 0.5 < t ∧ 0.5. Thus μ(x) < t or μ(y) < t, that is xt∈μ or yt∈μ. So xt∈ ∨qμ or yt∈ ∨qμ which is a contradiction. We know t ≤ 0.5, then μ(xy) ≥ t∧0.5 = t and so xy ∈ μt⊆ [μ]t. Similarly, we can prove that xay ∈ μt⊆ [μ]tfor all x, y ∈ [μ]tand a ∈ S. Conversely, let [μ]tbe a bi-ideal of S for all t ∈ (0, 0.5]. Let x, y ∈ S such that μ(xy) < μ(x) ∧ μ(y) ∧ 0.5, then there exists t ∈ (0, 0.5) such that μ(xy) < t < μ(x) ∧ μ(y) ∧ 0.5. Then x, y ∈ U(μ; t) ⊆ [μ]t, which implies xy ∈ [μ]t. Hence μ(xy) ≥ t or μ(x + y) + t > 1, which is a contradiction. Therefore μ(xy) ≥ μ(x) ∧ μ(y) ∧ 0.5. Similarly we can show that μ(xy) ≥ μ(x) ∧ μ(y) ∧ 0.5 for all a, x, y ∈ S. This completes the proof. ¤
Theorem 4.8. Let μ be a proper (∈, ∈ ∨q)-fuzzy bi-ideal of S such that |{μ(x)|μ(x) < 0.5}| ≥ 2. Then there exist two proper non-equivalent (∈, ∈ ∨q)-fuzzy bi-ideals of S such that μ can be expressed as the union of them.
Proof. Let {μ(x)|μ(x) < 0.5} = {t1, t2, . . . , tr}, where t1 > t2 > . . . > tr and r ≥ 4. Then the chain of (∈ ∨q)-level bi-ideals of μ is
[μ]0.5⊆ [μ]t1 ⊆ [μ]t2 ⊆ . . . ⊆ [μ]tr = S.
Let B and C be fuzzy sets in S defined by
ν(x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ t1 if x ∈ [μ]t1, t2 if x ∈ [μ]t2\ [μ]t1 .. . tr if x ∈ [μ]tr\ [μ]tr−1
and θ(x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ μ(x) if x ∈ [μ]0.5, k if x ∈ [μ]t 2\ [μ]0.5 t3 if x ∈ [μ]t3\ [μ]t2 .. . tr if x ∈ [μ]tr\ [μ]tr−1
respectively, where t3< k < t2. Then ν and θ are (∈, ∈ ∨q)-fuzzy bi-ideals of S, and ν, θ ≤ μ. The chains of (∈ ∨q)-level bi-ideals of ν and θ are, respectively, given by
[μ]t1 ⊆ [μ]t2 ⊆ . . . ⊆ [μ]tr
and
[μ]0.5⊆ [μ]t2⊆ . . . ⊆ [μ]tr.
Therefore ν and θ are non-equivalent and clearly μ = ν ∪ θ. This completes the proof. ¤
5. Implication-Based Fuzzy Bi-Ideals
Set theoretic multi-valued logic is a special case of fuzzy logic such that the truth values are linguistic variables (or terms of the linguistic variables truth). By using extension principle some operator like ∧, ∨, ¬, → can be applied in fuzzy logic. In fuzzy logic, [P ] means the truth value of fuzzy proposition P . In the following, we show a correspondence between fuzzy logic and set-theoretical notions.
[x ∈ A] = A(x), [x 6∈ A] = 1 − A(x), [P ∧ Q] = min{[P ], [Q]}, [P ∨ Q] = max{[P ], [Q]}, [P → Q] = min{1, 1 − [P ] + [Q]},
[∀x P (x)] = inf[P (x)], |= P if and only if [P ] = 1 for all valuations. We show some of important implication operators, where α denotes the degree of membership of the premise and β is the degree of membership of the consequence and I is the resulting degree of truth for the implication.
Early Zadeh Im(α, β) = max{1 − α, min{α, β}}, Lukasiewicz Ia(α, β) = min{1, 1 − α + β}, Standard Star (Godel) Ig(α, β) =
½ 1 if α ≤ β β otherwise Contraposition of Godel Icg(α, β) = ½ 1 α ≤ β 1 − α otherwise Gaines-Rescher Igr(α, β) = ½ 1 α ≤ β 0 otherwise Kleene-Dienes Ib(α, β) = max{1 − α, β}.
Definition 5.1. A fuzzy set μ of S is called an fuzzifying bi-ideal of S if it satisfies the following conditions:
(i) for any x, y ∈ S, ² [{[x ∈ μ] ∧ [y ∈ μ]} −→ [xy ∈ μ]], (ii) for any x, y, a ∈ S, ² [{[x ∈ μ] ∧ [y ∈ μ]} −→ [xay ∈ μ]].
Obviously the Definitions 5.1 and 2.1 are equivalent. Therefore there is no difference between fuzzifying bi-ideals and ordinary fuzzy bi-ideals.
Now, we have the concept of t−tautology, in fact ²tP if and only if [P ] ≥ t, for all valuations. Based on [29], we can extend the concept of implication-based fuzzy bi-ideals.
Definition 5.2. Let μ be an fuzzy set of S and t ∈ (0, 1]. Then μ is called a t−implication-based fuzzy bi-ideal of S if it satisfied the following conditions:
(i) for any x, y ∈ S, ²t[{[x ∈ μ] ∧ [y ∈ μ]} −→ [xy ∈ μ]], (ii) for any x, y, a ∈ S, ²t[{[x ∈ μ] ∧ [y ∈ μ]} −→ [xay ∈ μ]].
Corollary 5.3. A fuzzy set μ of S is a t−implication-based fuzzy bi-ideal of S if for all a, x, y ∈ S we have:
(i) I(μ(x) ∧ μ(y) ∧ μ(xy)) ≥ t, (ii) I(μ(x) ∧ μ(y) ∧ μ(xay)) ≥ t, where I is an implicative operator.
Proof. The result is clear by considering the Definitions 5.1 and 5.2. ¤ Theorem 5.4. (i) Let I = Igr(Gaines -Rescher.) Then μ is an 0.5−implication-based fuzzy bi-ideal of S if and only if μ is an fuzzy bi-ideal with thresholds (r = 0, s = 1) of S.
(ii) Let I = Ig (Godel). Then μ is an 0.5−implication-based fuzzy bi-ideal of S if and only if μ is an fuzzy bi-ideal with thresholds (r = 0, s = 0.5) of S.
(iii) Let I = Icg (Contraposition of Godel). Then μ is an 0.5−implication-based fuzzy bi-ideal of S if and only if μ is an fuzzy bi-ideal with thresholds (r = 0.5, s = 1) of S.
Proof. The proof is straightforward by considering the Definitions 5.1 and 5.2 and the definitions of implication operators. ¤
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