7fuzzy-caga-anan.pmd Fuzzy on Ideal Sets and a Fuzzy on Ideal Hahn-Banach Theorem 70 SCIENCE DILIMAN (JULY-DECEMBER 2018) 30:2, 70-86 Fuzzy on Ideal Sets and a Fuzzy on Ideal Hahn-Banach Theorem Lezel N. Mernilo-Tutanes Mathematics Department Bukidnon State University Randy L. Caga-anan* Department of Mathematics and Statistics MSU-Iligan Institute of Technology and PRISM ABSTRACT In set theory, an ideal is a collection of sets that are considered to be small or negligible, such that every subset of an element of the ideal m u s t a l s o b e i n t h e i d e a l a n d t h e u n i o n o f a n y t w o e l e m e n t s o f t h e ideal must also be in the ideal. A fuzzy set is a class of objects with grades of membership in the interval [0, 1]. It is used to mathematically represent uncertainty and provide a formal tool to deal with imprecisions present in many problems. We use ideals to def ine fuzzy on ideal sets, which can be seen as a generalization of the fuzzy sets. We establish s o m e o f i t s b a s i c p r o p e r t i e s , a n d w e s t a t e a n d p r o v e a H a h n - B a n a c h T h e o r e m w i t h t h e f u z z y o n i d e a l s e t s , w h i c h c a n b e s e e n a s a g e n e r a l i z a t i o n o f a f u z z y H a h n - B a n a c h T h e o r e m , w h i c h i n t u r n , i s a fuzzif ied generalization of an analytic form of the classical Hahn-Banach Theorem. Mathematics Subject Classification (2010): 62B86 Keyword s: Fuzzy, ideal, Hahn-Banach theorem _______________ *Corresponding Author ISSN 0115-7809 Print / ISSN 2012-0818 Online INTRODUCTION T h e co n ce p t of i d e a l s p a ce s w a s f i r s t s t u d i ed by Ku r a tow s k i ( 1 9 6 6 ) a n d Vaidyanathaswamy (1945). Formally, given a set X, an ideal I(X) is a nonempty collection of subsets of X that satisf ies: i. A ∈ I (X) and B ⊆ A implies B ∈ I (X); and ii. A ∈ I (X) and B ∈ I (X) implies A ∪ B ∈ I (X). L.N. Mernilo-Tutanes and R.L. Caga-anan 71 Ideal spaces were then imported to topology. For instance, Jankovic and Hamlett (1990) investigated the notion of topological spaces with ideals. Thereafter, ideal spaces found their way to other concepts in topology. In 1965, fuzzy sets were introduced by Zadeh (1965) and Klaua (1965) as extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms: an element either belongs or does not belong to the set. That is, a set A on a universal set X can be identif ied as the characteristic function of A having only the values 0 or 1. On the other hand, a fuzzy set on X is formally def ined as a mapping from X into the unit interval [0, 1]. Then in 1978, Zadeh introduced possibility theory as an extension of his theory of fuzzy sets and fuzzy logic (Zadeh 1978). Possibility theory should not be confused with probability theory. It is an uncertainty theory trying to make sense of incomplete information and is viewed as a complement to probability theory. Similar to a probability distribution, the theory uses a possibility distribution. A possibility distribution is a mapping π x from a set of states to a totally ordered set such as the unit interval [0, 1]. As one may easily notice, a possibility distribution can be used as an interpretation of the fuzzy sets. We provide the example given by Zadeh (1978) to better understand a possibility distribution and differentiate it from a probability distribution. Suppose we have the statement “Hans ate x eggs for breakfast,” with x taking values in U = {1,2,3,...}. We may associate a possibility distribution with x by interpreting π x (u) as the “degree of ease with which Hans can eat u eggs”. We may also associate a probability distribution with x by interpreting P x (u) as the probability of “Hans eating u eggs for breakfast”. The values of π x (u) and P x (u) may look like as shown in the following table. u 1 2 3 4 5 6 7 8 π x (u) 1 1 1 1 0.7 0.5 0.4 0.2 P x ( u ) 0.2 0.7 0.1 0 0 0 0 0 One may easily notice that the sum of all values for P x (u) is equal to 1, but for π x (u), it is not. We may also observe that the possibility that Hans may eat 3 eggs for breakfast is 1 but the probability that he may do so is quite small. Hence, we can say that a high degree of possibility does not imply a high degree of probability, nor does a low degree of probability imply a low degree of possibility. However, if an event is impossible, then it should be improbable. Fuzzy on Ideal Sets and a Fuzzy on Ideal Hahn-Banach Theorem 72 In this study, we want to capture statements like the following. Suppose we want to complete the statement “Tomorrow , ” we may have the following choices: a. will be sunny b. there will be rain c. will be cloudy d. there will be a storm We may give a value of 0.8 for the possibility that “tomorrow will be sunny”, 0.6 for the possibility that “tomorrow there will be rain”, 0.4 for the possibility that “tomorrow will be cloudy”, and 0.2 for the possibility that “tomorrow there will be a storm”. Now, consider the possibility that “tomorrow will be sunny and there will also be rain.” This condition is very rare but not impossible. We cannot just give this possibility a value equal to the inf imum of the possibility that “tomorrow will be sunny” and the possibility that “tomorrow there will be rain”. We know that this possibility should be far less than any of these two possibilities. Hence, we may give it a separate possibility value not dependent on the other two mentioned possibilities, say, 0.01. With this situation in mind, we def ine a fuzzy on ideal set in the next section. As an application of this new concept, we consider the Hahn-Banach Theorem. The theorem is no doubt an important and powerful result in functional analysis. It was generalized in many directions. One of it is by fuzzy sets, like what Rhie and Hwang (1999) did. In a similar way, we generalize the theorem with fuzzy on ideal sets. FUZZY ON IDEAL SETS We now formally def ine a fuzzy on ideal set. Def inition 2.1. Let X be a nonempty set I(X) an ideal on X, and I be the unit interval [0, 1]. A function μ:I(X)→I called a fuzzy on ideal set provided: i. μ (∅) = 0 and ii. for nonempty sets A, B ∈ I (X), with A ⊆ B, we have μ (B) ≤ μ (A). L.N. Mernilo-Tutanes and R.L. Caga-anan 73 We denote by I I(X) the set of all such μ. It is important to note here that the “reverse inequality”, μ (B) ≤ μ (A), of Def inition 2.1 encapsulates our preceding idea that the possibility that “tomorrow will be sunny and there will also be rain” should not just be equal to the inf imum of the possibility that “tomorrow will be sunny” and the possibility that “tomorrow there will be rain” as it can be far less. Remark 2.2. We may think of a fuzzy on ideal set as a generalization of a fuzzy set in X in the sense that the set of all fuzzy sets I X can be embedded in I P(X), where P(X) is the power set of X— the largest ideal on X . To see this, let μ : X → I be a fuzzy set in X. We can identify μ with μ ∈IP(X) def ined by Example 2.3. Let X = {a,b} and μ = {(a,1), (b,0.5)} be on X. We may identify μ with the fuzzy on ideal set μ def ined as μ = {({a},1), ({b},0.5), ({a,b},0), (∅,0)}. Example 2.4. Let X be a nonempty set and μ : X → I be a fuzzy set. We can def ine a fuzzy on ideal set π : P(X)→ I as π (A) = inf x∈A μ(x) and π (∅) = 0. This is called a guaranteed possibility in Dubois and Prade (2000). Remark 2.5. It is important to note that Def inition 2.1 does not def ine a measure. For A ⊆ B, a measure m should have m(A)≤m(B), not the reverse inequality, as in our def inition. Next, we def ine some relational operators between fuzzy on ideal sets. Def inition 2.6. Let X be a nonempty set and μ1, μ2 ∈ I I(X). We say that μ1 ≤ μ2 , μ1 ≥ μ2 or μ1 = μ2 , provided that, for every A∈ I(X), we have μ1(A) ≤ μ2(A), μ1(A) ≥ μ2 (A), or μ1(A) =μ2(A), respectively. Definition 2.7. Let X be a nonempty set and μ ∈ I I(X). The complement of μ denoted by μ c: I (X) →I is def ined by μ c (∅) = 0, and for ∅ ≠ A∈ I(X) , μ c : (A) = inf {1– μ ({x})}. Remark 2.8. For A = {x}, the preceding def inition coincides with the def inition of the complement of a fuzzy set. Def inition 2.9. Let X be a nonempty set and I(X) be an ideal on X. If {μ j | j∈ J} is a collection of fuzzy on ideal sets, then the union and the intersection of the μ j ’s are given by: ∼ ∼ μ (x), if A={x}, x∈X; ∼μ (A)= 0, otherwise. ∼ x∈A Fuzzy on Ideal Sets and a Fuzzy on Ideal Hahn-Banach Theorem 74 i. (∨ j∈J μ j )(A) = sup {μ j (A)| j ∈ J}; and ii. (∧ j∈J μ j )(A) = inf {μ j (A)| j ∈ J}, respectively, for every A ∈ I(X). Next, we show that the complement, union, and intersection of fuzzy on ideal sets are also fuzzy on ideal sets. Proposition 2.10. Let X be a nonempty set and I(X) be an ideal on X. If {μ j | j ∈ J} is a collection of fuzzy on ideal sets, then μ c j , ∨ j∈J μ j , and ∧ j∈J μ j are fuzzy on ideal sets. Proof. Let ∅ ≠ A, B ∈ I(X), such that A ⊆ B. Then, {μ ({x}) : x∈A} ⊆{μ ({x}) : x ∈ B}, a n d s o , { 1 – μ ( { x } ) : x ∈ A } ⊆ { 1 – μ ( { x } ) : x ∈ B } . T h u s , f o r j ∈ J , μ c j ( A ) = inf{1–μ j ({x})} ≥ inf{1–μ j ({x})}= μ c j ( B). By Def inition 2.7, μ c j (∅) = 0 for each j∈J, and hence, μ c j is a fuzzy on ideal set. Now, since A ⊆ B and μ j ∈ II(X), μ j (A) ≥ μ j (B) where j∈J. It follows that sup{μ j (A)| j ∈ J}≥ sup{μ j (B)| j ∈ J}, and so, (∨ j∈J μ j )(A) ≥ (∨ j∈J μ j )(B). Note that (∨ j∈J μ i )(∅) = sup{μ j (∅) | j ∈ J}= 0, and hence, ∨ j∈J μ j is a fuzzy on ideal set. Similarly, we can show that ∧ j∈J μ j is a fuzzy on ideal set.  MAPPINGS In this section, we show that, given only a mapping between nonempty sets (not a mapping between fuzzy on ideals sets), we can def ine the image and pre-image of fuzzy on ideal sets. The following are preparatory def initions and results. Definition 3.1. Let X and Y be nonempty sets, and let f : X → Y be a mapping. If I(X) and I(Y) are ideals on X and Y, respectively, we def ine f (I(X)) = {f (A) : A ∈ I (X)} and f –1(I(Y)) = {A : A ⊆ f –1 (B), B ∈ I (Y )}, where f (A) and f –1 (B) are the usual image and pre-image of A ⊆ X and B ⊆ Y, respectively. Theorem 3.2. Let X and Y be nonempty sets and let f : X → Y be a mapping. If I(X) and I(Y) are ideals on X and Y, respectively, then f(I(X)) and f –1(I(Y)) are ideals on Y and X, respectively. Proof. Since I(X) is an ideal on X, I(X)≠∅. It follows that f(I(X)) ≠∅. Let B 2 ∈f (I(X)). Then, there exists A 2 ∈ I (X), such that f (A 2 ) = B 2 . Let B 1 ⊆ B 2 . Now, let A 1 = A 2 ∩ f –1 (B 1 ). Then, A 1 ⊆ A 2 and f (A 1 ) = B 1 . Note that A 1 ∈ I (X), since it is a subset of A 2 ∈ I (X). Thus, x∈Bx∈A L.N. Mernilo-Tutanes and R.L. Caga-anan 75 B 1 ∈ f (I (X)). Next, suppose that B 1 , B 2 ∈ f (I (X)), then there exist A 1 , A 2 ∈ I (X), such that f (A 1 ) = B 1 and f (A 2 ) = B 2 . Now, A 1 ∪ A 2 ∈ I (X), since I (X) is an ideal. Thus, B 1 ∪ B 2 = f (A 1 ) ∪ f (A 2 ) = f (A 1 ∪ A 2 )∈ f (I (X)). Therefore, f (I (X)) is an ideal on Y. Since I (Y) is an ideal on Y, I (Y) ≠ ∅. It follows that f –1 (I(Y)) ≠ ∅. Let A 2 ∈ f –1(I(Y)). Then, there exists B 2 ∈ I(Y) such that A 2 ⊆ f –1(B 2 ) . Let A 1 ⊆ A 2 . Then, we also have A 1 ⊆ f –1(B 2 ), a n d t h u s , A 1 ∈ f –1(I(Y)). N e x t , s u p p o s e t h a t A 1 , A 2 ∈ f –1(I(Y)), t h e n t h e r e e x i s t B 1 , B 2 ∈ I(Y ), such that A 1 ⊆ f –1(B 1 ) and A 1 ⊆ f –1(B 2 ). Since B 1 ∪B 2 ∈ I(Y), I(Y) being an ideal on Y, we have A 1 ∪A 2 ⊆ f –1(B 1 ∪B 2 ) , then A 1 ∪A 2 ⊆ f –1 (I(Y)). Thus, f –1 (I(Y)) is an ideal on X .  The next def inition def ines the image and pre-image of fuzzy on ideal sets out of an ordinary mapping, and the next proposition proves that it is well-def ined. Definition 3.3. Let X and Y be nonempty sets, and let f : X → Y be a mapping. Let μ ∈ II(X) and σ ∈ II(Y) for some ideals I (X) and I (Y) of X and Y, respectively. Def ine the image of μ, denoted by f [μ], and the pre-image of σ, denoted by f –1[σ], as follows: i. f [μ] : f (I(X)) →I , where for B ∈ f (I(X)), f [μ](B) = sup μ (A), where S B = {A∈ I(X)) : f (A) = B}; and ii. f –1[σ ]:f –1(I(Y))→I, where for A ∈ f –1(I(Y)), f –1[σ ] (A) = (σ ° f )(A). Proposition 3.4. Let X and Y be nonempty sets, and let f : X → Y be a mapping. Let μ ∈ I I(X) and σ ∈ II(Y). Then, f [μ] and f –1[σ ] are fuzzy on ideal sets on Y and X, respectively. Proof. We f irst note that, by Theorem 3.2, f (I(X)) and f –1(I(Y)) are ideals on Y and X respectively. If B ≠ ∅, then S B = {∅}, and so, f [μ]∅ = sup A∈ SB μ(A) = 0. Also, for ∅ = A ∈ f –1(I(Y)), we have f (A) = ∅, and so, f –1[σ]:f –1(∅) = σ (f(A)) = 0. Now, let ∅ ≠ B 1 , B 2 ∈ f (I(X)), such that B 1 ⊆ B 2 . Then, for A 1 , A 2 ∈I(X) , such that f (A 1 )=B 1 , f (A 2 )=B 2 , and A 1 ⊆ A 2 , we have μ (A 2 ) ≤ μ (A 1 ) , since μ is a fuzzy on ideal set. Hence, sup A∈SB 2 μ (A) ≤ sup A∈SB 1 μ (A), and so, f [μ](B 2 ) ≤ f [μ](B 1 ). Now, for ∅ ≠ A 1 , A 2 ∈ f –1(I(Y)), such that A 1 ⊆ A 2 , we have f (A 1 ) ⊆ (A 2 ) . Hence, σ f (A 2 )) ≤ σ f (A 1 )), since σ is a fuzzy on ideal set. Consequently, f –1[σ ](A 2 ) ≤ f –1[σ ](A 1 ). Therefore, f [μ] and f –1[σ ] are fuzzy on ideal sets on Y and X, respectively.  A∈SB Fuzzy on Ideal Sets and a Fuzzy on Ideal Hahn-Banach Theorem 76 FUZZY ON IDEAL HAHN-BANACH THEOREM Rhie and Hwang (1999) fuzzif ied the analytic form of the Hahn-Banach Theorem. They built the idea from the works of Katsaras (1981, 1984) and Katsaras and Liu (1977) on fuzzy vector spaces and fuzzy seminorm, and the work of Krishna and Sarma (1991) on the generation of the fuzzy vector topology from an ordinary vector topology. We state and prove the analytic form of the Hahn-Banach theorem in the fuzzy on ideal setting. This can be seen as a generalization of the fuzzy Hanh- Banach Theorem by Rhie and Hwang (1999). We follow the ideas and the flow of proof by Rhie and Hwang (1999). We recall f irst that for a vector space X over and A, B ⊆ X, we have A+B={a+b:a∈A and b∈B} and for , tA = {ta:a∈A}. Let I(X) be an ideal on X. We def ine an associated set X 0 by X 0 = {a:{a} ∈ I(X)}. That is, X 0 is the set out of the singleton subsets of I(X). Note that X 0 ⊆ X. For ∅ ≠ A∈I(X) , by the f irst property of an ideal, all singleton subsets of A are also in I(X), and so, A ⊆ X 0 . We want next that I(X) be closed under f inite addition and scalar multiplication. The next proposition shows that it is enough to assume that X 0 is a linear subspace of X , such that X 0 ∈ I (X). Proposition 4.1. Let X be a vector space over , and let X 0 be a linear subspace of X. If I (X) is an ideal of X, such that X 0 ∈ I (X), then for every A, B ⊆ X 0 and every , we have tA, A+B∈I(X). Proof. Let A, B ⊆ X 0 and . Since X 0 is a linear subspace of X, it follows that tA = {ta : a∈A} ⊆ X 0 and A+B ={a+b : a∈A and b∈B} ⊆ X 0 . Since X 0 ∈ I(X), then any subset of X 0 is in I(X). That is, tA and A+B are in I(X) .  To move forward we need to def ine f inite addition and scalar multiplication of fuzzy on ideal sets, such that the result is also a fuzzy on ideal set. Def inition 4.2. Let X be a vector space over , and I(X) be an ideal on X , such that X 0 is a linear subspace of X and X 0 ∈ I(X). For any μ, v ∈ II(X), we def ine μ+v as follows: (μ+v)(∅)=0 and for ∅ ≠ A∈I(X),             ngleton)s not a si (or A iotherwise.Axxv n)a singleto (or A is A={x}; ifXxxxvx Av xxx },:}))({{(inf },,:})({})({{sup =))(( 02121 =21      ∈ ∈ ∈ L.N. Mernilo-Tutanes and R.L. Caga-anan 77 In the case A = {x} in Def inition 4.2, we note that, since X 0 is a linear subspace of X, we can always express x as x = 0 + x, and thus, the set under the sup is never empty. Def inition 4.3. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈ I(X). For a scalar and ∈ II(X) , we def ine tμ as follows: (tμ)(∅) = 0 and for ∅ ≠ A∈I(X), ∈             A = {0 } .i f t= 0 a n dXyysu p { 0 } ; Aif t = 0 a n d 0 ;if tAt At ,:} )({ 0 , , =))(( 0 1    Proposition 4.4. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈I(X). If and μ, v ∈ II(X), then tμ and μ+v are in I I(X). Proof. We note that, by Def inition 4.3, we have (tμ)(∅) = 0. Now, let ∅ ≠ A 1 , A 2 ∈I(X), such that A 1 ⊆ A 2 and . Let t ≠ 0. Then, and t-1 A 1 ⊆ t-1A 2 . Note that t–1A 1 , A 2 ∈I(X) by Proposition 4.1. It follows that μ (t–1A 1 ) ≥ μ (t–1A 2 ) since μ ∈ II(X). Suppose t = 0 and A 1 , A 2 ≠ {0}, we have (tμ)(A 1 ) = 0 = (tμ)(A 2 ). Now, for t = 0 and A 1 , A 2 = {0}, we have (tμ)(A 1 ) = (tμ)(A 2 ) . If t = 0, A 1 = {0} and A 2 ≠ {0} , then (tμ)(A 2 ) = 0 ≤ (tμ)(A 1 ). Thus, tμ ∈ II(X). Next, we show that μ + v ∈II(X). By Def inition 4.2, (μ + v)(∅)=0. Let ∅≠ A 1 , A 2 ∈I(X), such that A 1 ⊆ A 2 . Suppose A 2 is a singleton, then A 1 must be a singleton. Then, A 1 = {x}=A 2 . Notice that, (μ + v)(A 1 ) = (μ + v)(A 2 ). Now, if A 2 is not a singleton, then A 1 is either a singleton or not. In any case, since A 1 ⊆ A 2 , we have{(μ+v)({x}): x∈A 1 }⊆{(μ+v )({x}): x∈A 2 }. Thus, inf{(μ+v)({x}): x∈A 1 } ≥ inf{(μ+v)({x}): x∈A 2 }. Consequently, (μ+v)(A 1 ) ≥ (μ+v)(A 2 ). Therefore, μ + v ∈ I I(X). Next, we need to def ine a fuzzy on ideal seminorm. We begin by def ining its properties. ∈ ∈   Fuzzy on Ideal Sets and a Fuzzy on Ideal Hahn-Banach Theorem 78 Def inition 4.5. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈ I(X). A ρ ∈ I I(X) is said to be i. convex if ρ (tA+(1–t)B) ≥ min{ρ (A), ρ (B)} for every t∈ [0,1] and ∅ ≠ A, B ∈ I(X); ii. balanced if (tρ)(A) ≤ ρ (A) for every with | t |≤ 1 and ∅ ≠ A ∈ I(X); iii. absorbing if sup 1>0 (tρ)(A)=1 for every ∅ ≠ A ∈ I(X). Def inition 4.6. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈I(X). A ρ ∈ I I(X) is called a fuzzy on ideal seminorm if it is convex, balanced, and absorbing. Associated with a fuzzy on ideal seminorm, we def ine below an important mapping, and then we prove that it has the properties of an ordinary seminorm. Def inition 4.7. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈I(X) . Let ρ be a fuzzy on ideal seminorm. For each ε ∈ (0,1), we def ine Pε : I (X) →[ 0 , +∞) by Pε (∅) = 0 and for ∅ ≠ A ∈ I( X), Pε (A)=inf{t>0:(tρ)(A)>ε}. Observe that Pε is well-def ined since ρ is absorbing. Moreover, 1= sup1>0 (tρ)({0}) = sup 1>0 (ρ)(t-1{0})=sup 1>0 (ρ)({0}) implies that ρ ({0})=1. Remark 4.8. For 0< ε 1 < ε 2 < 1 , we have {t>0:( t ρ (A) ε 2 }⊆{t >0:(t ρ) (A) > ε 1 }. Hence, Pε (A)=inf{t>0 : (tρ)(A)>ε 1}≤ inf{t>0 : (tρ)(A)>ε 2}=Pε 2 (A). That is, {Pε} is increasing in ε . Theorem 4.9. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈I(X). If ρ is a fuzzy on ideal seminorm, then, for each ε ∈ (0,1), we have Pε satisfying the following properties: i. , for all and ∅ ≠ A ∈ I(X) ; ii. , for ∅ ≠ A,B ∈ I(X) . ∈ ( ) =| | ( )P A P A   ( ) ( ) ( )P A B P A P B     L.N. Mernilo-Tutanes and R.L. Caga-anan 79 Proof. i. Let ε ∈(0,1) and ∅ ≠ A ∈ I(X). If α = 0, then αA = {0}. Observe that Let α ≠ 0. Since ρ is balanced, (tρ )(A) ≤ ρ (A) for | t |≤1. In particular, t = –1 implies (–ρ)(A) ≤ ρ (A), and so, . Now, Hence, . Next, consider that . It follows that . Thus, . Consequently, Then, ii. Let ∅ ≠ A, B ∈ I( X), r ∈{t >0 : (tρ)(A) >ε}, and s ∈{t >0 : (tρ)(B) >ε}. Then, (rρ)(A) >ε and (sρ)(B) >ε. Now, by convexity of ρ, we have . It follows that and so, . . Hence, we have 0 ∙ 0: 0 0: 1 0 0: 0 0:1 0 |0| . 1 1 1 . 0: 0: 0: | | | | ′ 0: ′ , | | ′ 0: ′ | | ′ 0: ′ | | 0: | | . by the preceding result where ′ | | 0: 0: ⊆ 0: . , , , ∈ 0: . Fuzzy on Ideal Sets and a Fuzzy on Ideal Hahn-Banach Theorem 80 Thus, Therefore, .  The next theorem shows that the inf imum of the Pε has properties similar to it. The orem 4.10. Let X be a vector space over , and I( X ) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈ I(X). Let ρ be a fuzzy on ideal seminorm. Then, the function P:I (X) → [0,+∞) def ined by P(∅)=0, and for ∅ ≠ A ∈ I( X), P(A) = inf{Pε (A):ε ∈(0,1)} satisf ies the following properties: i . for all and ii. , for all . Proof. The f irst property follows directly from the f irst proper ty in Theorem 4.9. Let ∅ ≠ A,B ∈I( X). Since {P ε} is increasing in ε, for every ∅ ≠ A∈I(X ), P(A) = inf{Pε (A):ε ∈(0,1)} = limε→0 Pε (A). Thus, P(A+B)=inf{Pε (A+B):ε ∈(0,1)}≤ inf{Pε (A)+Pε (B):ε ∈(0,1)} = lim{Pε (A)+Pε(B)} = lim{Pε (A)+lim Pε(B)=P (A)+P (B).  The next two theorems give us the relationship between fuzzy on ideal seminorms and its associated mappings having the ordinary seminorm properties. It is our key to tap on the classical Hahn-Banach Theorem that will be used in the proof of our fuzzy on ideal Hahn-Banach Theorem. Theorem 4.11. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈ I(X). Furthermore, let ρ 1 and ρ 2 be two fuzzy on ideal seminorms. If for every A∈I( X) , ρ 1 (A) ≤ ρ 2 (A), then for every ε ∈(0,1), P1(A) ≥ P 2(A) for all A∈I( X). Proof. If for every A∈I (X), ρ 1 (A) ≤ ρ 2 (A), then for every A∈I(X) and t > 0, (tρ 1 )(A) = ρ 1 (t-1A) ≤ ρ 2 (t-1A) = (tρ 2 )(A). Let ε ∈(0,1) and ∅ ≠ A∈I(X). Observe that {t>0:(tρ 1 )(A) >ε} is a subset of {t > 0:(tρ 2 )(A)>ε}. Hence, inf{t > 0:(tρ 1 )(A) >ε} ≥ {t > 0:(tρ 2 )(A)>ε}. Thus, P1(A) ≥ P2(A).  0: 0: 0: . | | , ∈ ∅ ∈ ; ∅ , ∈ ε ε ε→0 ε→0ε→0 ε ε L.N. Mernilo-Tutanes and R.L. Caga-anan 81 Remark 4.12. The converse of Theorem 4.11 does not always hold. To see this, let X = and I(X) = P( ). Def ine ρ 1 and ρ 2 as follows: One can check that ρ 1 and ρ 2 are fuzzy on ideal seminorms and for ε ∈ (0,1), but ρ 1 ≤ ρ 2 and ρ 2 ≤ ρ 1 . The following *-property will give us a suff icient condition for the converse to hold. Def inition 4.13. Let X be a vector space over , and let I(X) be an ideal on X, such that X 0 is a linear subspace of X. Let X 0 ∈ I(X). Let ρ be a fuzzy on ideal seminorm. We say that ρ has the *-property if, for every ∅≠A∈ I(X), we have ρ (A) = inf{ρ (tA):0 < t < 1}. An example of a fuzzy on ideal seminorm with the *-property will be given later. It is a crucial part of our main theorem. In the meantime, let us prove that, with the *-property, the converse of Theorem 4.11 will hold. Lemma 4.14. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X. Let X 0 ∈ I(X). Let ρ be a fuzzy on ideal seminorm with the *-property. If ∅≠A∈ I(X) and ρ (A)<ε <1, then Pε (A)>1. Proof. Let ∅≠A∈ I(X) and ρ (A)<ε <1. Since ρ is balanced being a fuzzy on ideal s e m i n o r m , ( t ρ ) ( A ) ≤ ρ ( A ) < ε f o r | t | ≤ 1 . T h u s , Pε ( A ) = i n f { t > 0 : ( t ρ ) ( A ) > ε } = inf{t>1:(tρ)(A)> ε}≥1. We are left to show that Pε (A)≠1. Suppose Pε (A)=1. Then, (tρ)(A)>ε for all t >1. S i n c e ρ h a s t h e *- p r o p e r t y, ρ ( A ) = i n f { ρ ( t A ) : 0 < t < 1 } = i n f { ( t - 1 ρ ) ( A ) : 0 < t < 1 } =inf{(tρ)(A):t >1}≥ε. However, this is a contradiction, since ρ (A) < ε. Therefore, Pε (A)>1.  2 1, ∈ 1,1 ; 1 3 , ∈ 5, 1 ∪ 1,5 ; 0, . 1 1, ∈ 1,1 ; 1 3 , ∈ 5, 1 ∪ 1,5 ; 0, . 1 2 | | Fuzzy on Ideal Sets and a Fuzzy on Ideal Hahn-Banach Theorem 82 Theorem 4.15. Let X be a vector space over , and let I(X) be an ideal on X, such that X 0 is a linear subspace of X. Let X 0 ∈ I(X). Let ρ 1 and ρ 2 be two fuzzy on ideal seminorms, with ρ 2 having the *-property. If for every ε ∈ (0,1), we have for every A∈ I(X), then for every A∈ I(X). Proof. If A = ∅, then by def inition, . Suppose that for every ε ∈ (0,1), , and that there exists a B∈ I(X), such that ρ 2 (B) < ρ 1 (B) . Let ρ 2 (B) < ε < ρ 1 (B). If t = 1, then tρ 1 (B) = ρ 1 (B) > ε, and so, . Since ρ 2 is balanced, . By Lemma 4.14, . Thus, . This is a contradiction to our assumption that for every ε ∈ (0,1) , , . Therefore, we must have , .  We now def ine and prove an important fuzzy on ideal seminorm with the *-property. We begin with the following def inition. Def inition 4.16. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈ I(X). Let M be also a linear subspace of X and f:M→ be a linear functional. Furthermore, let I(M) be an ideal of M, such that M 0 ∈ I(M) . We associate with f the function def ined by , and for , . Let We def ine by χ Bf (∅) = 0, and for ∅ ≠A∈ I(X), Observe that, for t > 0, . For convenience, whenever we have , we let the and say that the . Theorem 4.17. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈ I(X). Let M be also a linear subspace of X and f:M→ be a linear functional. Furthermore, let I(M) be an ideal of M, such that M 0 ∈I(M) . Then, χ Bf is a fuzzy on ideal seminorm with the *-property . Proof. We f irst show that χ Bf has the *-proper ty. Let ∅≠A∈ I(M). Suppose that Then, . Then, for all 0 < t < 1, sup x∈Α|f(tx)|= t sup x∈Α|f(x)|≤ t<1. Hence, χ Bf (tA) = 1, for all 0 < t < 1, and so, inf{χBf(tA):0 < t < 1}= 1. Suppose that χ Bf (A) = 0. Then, π f (A) = sup x∈Α|f (x)| >1. Assume inf{χBf(tA)|0 < t < 1}= 1. It follows that, for all 0)(:0>{=)( 1 1  BttinfBP   1}>)(:0>{=)( 2 2  BttinfBP   1>)(2 BP 2 1( ) > ( )P B P B    )()( 21 APAP     )( XIA    )( XIA   )()( 21 AA   : → ∪ ∞   0=)(f   )(MIA   |)(|sup=)( xfA Axf  ∈ : ∅ : → 0,1   )()( AttA ff      |)(|sup=)( xfA Axf 1. ∈ | | 1 lim t→+∞πf (A)= 1 πf (A) =1t t 1 t ∈ | | 1 1 . 1, ∈ ; 0, . 0 0 1 L.N. Mernilo-Tutanes and R.L. Caga-anan 83 for all η >0. Since η is arbitrary, sup x∈Α|f(x)| ≤ 1, which contradicts to χBf(A) = 0. Hence, there exists 0 < t < 1, such that χ Bf (tA) = 0, that is inf{χ Bf (tA):0 < t < 1}= 0. Therefore, χ Bf has the *-property. Next we show that it is a fuzzy on ideal set. It is enough to show the reverse inequality. Let ∅ ≠ A,B ∈ I(M), such that A ⊆ B. If χχ Bf (B) = 0, then the reverse inequality automatically follows. Suppose that χ Bf (B) = 1. Then, π f (B) = sup x∈Β | f (x)| ≤1. Now, π f (A) = sup x∈Α | f (x)| ≤ supx∈Β | f (x)| ≤ 1. Hence, χ Bf (A) = 1. Thus, χ Bf (B) ≤ χ Bf(A). We are left to show that χ Bf is convex, balanced, and absorbing. Let ∅ ≠ A,B ∈ I(M). If χ Bf (A) = 0 or χ Bf (B) = 0, then the inequality for convex is clearly satisf ied. Suppose that A,B ∈ B f , that is, χ Bf (A) = 1 and χ Bf(B) = 1. Then, s u p x ∈ Β | f ( x ) | ≤ 1 and supx∈Β |f (x)| ≤1. It follows that, for each 0 ≤ t ≤1, since f is linear, we have ≤ t + 1–t = 1. Thus, χBf (tA+(1–t)B) = 1, and so χBf is convex. Let ∅ ≠ A,B ∈ I(M) and | t| ≤ 1. If (tχχ Bf)(A) = 0 , then the inequality for balanced is clearly satisf ied. Suppose (t χ B f )( A) = 1 . Then for each t with | t| ≤ 1, we have . Hence, sup x∈Α| f(x)|≤| f(x)|≤|t| ≤ 1, and so, χχ Bf (A)= 1. Consequently, χ Bf is balanced. Lastly, let ∅ ≠ A ∈ I (M). If πf (A) = supx∈Α | f (x)| < +∞, t a k e t 0 = s u p x ∈ Α | f ( x ) | ≤ | f ( x ) | . T h e n , . T h u s , (t 0 χ Bf) (A)=1. Therefore, supt >0 (tχ Bf)(A)= 1. If πf (A) = supx∈Α|f (x)| ≤ +∞, we have remarked prior to this theorem that supt >0 (tχ Bf)(A)= 1. Thus, χBf is absorbing.  Finally, we have the fuzzy on ideal Hahn-Banach Theorem. Theorem 4.18. Let X be a vector space over , and I(X) be an ideal on X, such that X 0 is a linear subspace of X and X 0 ∈ I(X). Let M be also a linear subspace of X, and I(M) be an ideal of M, such that M 0 ∈ I(M) and M 0 ⊆ X 0 . Let ρ ∈ II(X) be a fuzzy on ideal seminorm. If f:M 0 → is a linear functional, such that χ Bf (A) ≥ ρ (A) for all A ∈ I (M), then there exists a linear functional g:X 0 → , such that: i. f(x) = g (x), x ∈M0; and ii. χ Bg (A) ≥ ρ (A) for all A∈I(X). ∈ 1 | | ∈ , ∈ | 1 | ∈ 1 ∈ | | 1 ∈ 0 1 0 ∈ | | 1 ∈ , ∈ | | 1 | | Fuzzy on Ideal Sets and a Fuzzy on Ideal Hahn-Banach Theorem 84 Proof. Let f be a linear functional on M 0 , such that χ Bf ( A) ≥ ρ (A) for all A ∈I (M) . Note f irst that χ Bf is a fuzzy on ideal seminorm with the *-property. In Theorem 4.11, let χ Bf=ρ2 and ρ=ρ1. Then, the corresponding Pε 2 is given by: for any ∅ ≠ A∈ I(M) and ε ∈(0,1), Pε 2(A) =inf{t>0:tρ 2 (A) >ε } =inf{t>0:ρ 2 (A/t) >ε } =inf{t>0:ρ 2 (A/t) =1}, since ρ 2 is a characteristic function =inf{t>0:π f (A/t) ≤1}, that is (A/t) ∈ B f =inf{t>0:π f (A) ≤t}, since f is linear =π f (A). Thus, by Theorem 4.11, for all ε ∈(0,1), Pε 2(A) = π f (A) ≤ Pε 1(A) for all ∅ ≠ A ∈ I(M), where Pε 1(A) =inf{t>0:tρ (A) >ε}for all ∅ ≠ A ∈ I(X). Observe that we are considering here A ∈ I(X), instead of just I(M). This can be done because ρ ∈ II(X ). By the last inequality, π f (A) = sup x∈Α | f (x)| ≤ P(A) = inf{Pε 1(A):ε ∈(0,1)} for all ∅ ≠ A ∈ I(M). In particular, |f (x)| ≤ P({x}) , for all x ∈ M 0 . Note that by Theorem 4.10, P restricted to the singletons can be seen as a sublinear functional on X 0 . Hence, by applying the classical Hahn-Banach Theorem, there exists a linear functional g:X 0 → such that: i. f(x) = g (x), x ∈ M0; and ii. |g (x)| ≤ P({x}), for all x ∈ X 0 . Now, let ∅ ≠ A ∈ I(X). Note f irst that, by the def inition of P and the reverse inequality satisf ied by ρ, if x ∈ A, then P ({x}) ≤ P(A). Hence by (ii),  x ∈ A, |g (x)| ≤ P(A). Thus, we have sup x∈Α |g(x)| = πg (A)≤ P(A) for all ∅ ≠ A ∈ I(X). Let χBg =ρ2 in Theorem 4.15. Then, the corresponding Pε 2 is given by: for any ∅≠A∈ I(X) and ε ∈(0,1), Pε 2(A) =inf{t>0:tρ 2 (A) >ε } =inf{t>0:ρ 2 (A/t) >ε } =inf{t>0:ρ 2 (A/t) =1}, since ρ 2 is a characteristic function =inf{t>0:π g (A/t) ≤1}, that is (A/t) ∈ B g =inf{t>0:π g (A) ≤t}, since g is linear =π g (A). L.N. Mernilo-Tutanes and R.L. Caga-anan 85 Thus, for all ε ∈(0,1), Pε 2(A) ≤ P(A) = inf{Pε 1(A):ε ∈(0,1)} for all ∅ ≠ A ∈ I(X). Hence, for all ε ∈(0,1), Pε 2(A) ≤ Pε 1(A) = inf{t>0:(tρ )(A) >ε } for all ∅ ≠ A ∈ I(X). Since χ B g has the *-proper ty, by Theorem 4.15, we have χ B g (A) ≥ ρ (A) for all A ∈ I(X).  ACKNOWLEDGEMENT The f irst author was supported by the Commission on Higher Education (CHED) of the Philippines and the second author by the Premier Research Institute of Science and Mathematics (PRISM) of MSU-IIT. REFERENCES Dubois D, Prade H. 2000. Fundamentals of fuzzy sets. New York: Springer US. J a n k o v i c D , H a m l e t t T R . 1 9 9 0 . N e w t o p o l o g i e s f r o m o l d v i a i d e a l s . A m e r i c a n Mathematical Monthly. 97(4):295-310. Katsaras AK. 1981. Fuzzy topological vector space I. Fuzzy Sets and Systems. 6(1):85-95. Katsaras AK. 1984. Fuzzy topological vector space II. Fuzzy Sets and Systems. 12(2):143- 154. Katsaras AK, Liu DB. 1977. Fuzzy vector spaces and fuzzy topological vector spaces. Journal of Mathematical Analysis and Applications. 58(1):135-146. Klaua D. 1965. Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsberichte der Königlichen Preussische Akademie des W issenschaften zu Berlin. 7:859-876. Krishna SV, Sarma KKM. 1991. Fuzzy topological vector spaces-topological generation and normability. Fuzzy Sets and Systems. 41(1):89-99. Kuratowski K. 1966. Topology. New York: Academic Press. Rhie GS, Hwang IA. 1999. On the Fuzzy Hahn-Banach Theorem-an analytic form. Fuzzy Sets and Systems. 108(1):117-121. Vaidyanathaswamy R. 1944. The localisation theory in set topology. Proceedings Indian Academy of Sciences (Mathematical Sciences) 20(1):51-61. Zadeh LA . 1965. Fuzzy sets. Information and Control. 8(3):338-353. Za d e h LA . 1 9 7 8 . F u z zy s e t s a s a b a s i s fo r a t h eo r y of p o s s i b i l i t y. F u z zy S e t s a n d Systems. 1:3-28. Fuzzy on Ideal Sets and a Fuzzy on Ideal Hahn-Banach Theorem 86 _____________ Randy L. Caga-anan, Ph.D. (randy.caga-anan@g.msuiit.edu.ph) is an Associate Professor at the Department of Mathematics and Statistics, Mindanao State University- Iligan Institute of Technology. He took his Ph.D. from the University of the Philippines Diliman, and is currently working on topology, graph theory, and partial differential equations. Lezel N. Mernilo-Tutanes, Ph.D. (l_mernilo@yahoo.com) is an Instructor at the Mathematics Department, Bukidnon State University. She took her Ph.D. at the Mindanao State University- Iligan Institute of Technology with a dissertation on topology.