#### Abstract

In this paper, we essentially deal with Köthe-Toeplitz duals of fuzzy level sets defined using a partial metric. Since the utilization of Zadeh’s extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct some classical notions. In this paper, we present the sets of bounded, convergent, and null series and the set of sequences of bounded variation of fuzzy level sets, based on the partial metric. We examine the relationships between these sets and their classical forms and give some properties including definitions, propositions, and various kinds of partial metric spaces of fuzzy level sets. Furthermore, we study some of their properties like completeness and duality. Finally, we obtain the Köthe-Toeplitz duals of fuzzy level sets with respect to the partial metric based on a partial ordering.

#### 1. Introduction

By , we denote the set of all sequences of fuzzy numbers. We define the classical sets , , and consisting of the sets of all bounded, convergent, and null series, respectively; that is We can show that , , and are complete metric spaces with the partial metric defined by where and are the elements of the sets , , or .

Secondly, we introduce the sets , , and consisting of sequences of -bounded variation by using the partial metric with respect to the partial ordering , as follows: where the distance function denotes the partial metric of fuzzy level sets defined by for any with the partial ordering . One can conclude that the sets , , and are complete metric spaces with the following partial metrics: respectively, where and are the elements of the sets , , or and for all .

Many authors have extensively developed the theory of the different sets of sequences and its matrix transformations [1, 2]. Following Başar [3, page 347], we note that Mursaleen and Basarır [4] have recently introduced some new sets of sequences of fuzzy numbers generated by a nonnegative regular matrix some of which reduced to the Maddox's spaces , , , and of sequences of fuzzy numbers for the special cases of that matrix . Quite recently, Talo and Başar [5] have extended the main results of Başar and Altay [6] to fuzzy numbers and defined the alpha-, beta-, and gamma-duals of a set of sequences of fuzzy numbers and gave the duals of the classical sets of sequences of fuzzy numbers together with the characterization of the classes of infinite matrices of fuzzy numbers transforming one of the classical set into another one. Also, Kadak and Başar [7–9] have recently studied fourier series of fuzzy valued functions and gave some properties of the level sets together with some inclusion relations, in [10]. Finally, Kadak and Ozluk [11–13] have introduced the sets , , , and of classical sequences of fuzzy level sets and sufficient conditions for partial completeness of these are established by means of fuzzy level sets.

The rest of this paper is organized as follows. In Section 2, some required definitions and consequences related with the partial metric and fuzzy level sets, sequences, and convergence are given. Section 3 is devoted to the completeness of the sets of sequences , , and , , of fuzzy level sets and some related notions. In the final section of the paper, the Köthe-Toeplitz duals of some classical sets are determined and given some properties including solidness.

#### 2. Preliminaries, Background and Notation

Motivated by experience from computer science, nonzero self-distance seen to be plausible for the subject of finite and infinite sequences.

*Definition 1 (see [14]). *Let be a nonempty set and be a function from to the set of nonnegative real numbers. Then the pair is called a* partial metric space* and is a* partial metric* for , if the following partial metric axioms are satisfied for all :(P1) if and only if ,(P2),(P3),(P4).

Proposition 2 (Nonzero self-distance [15]). *Let be the set of all infinite sequences over a set . For all such sequences and let , where is the largest number (possibly ) such that for each . Thus is defined to be over to the power of the length of the longest initial sequence common to both and . It can be shown that is a metric space.*

Each partial metric space thus gives rise to a metric space with the additional notion of nonzero self-distance introduced. Also, a partial metric space is a generalization of a metric space; indeed, if an axiom is imposed, then the above axioms reduce to their metric counterparts. Thus, a metric space can be defined to be a partial metric space in which each self-distance is zero.

It is clear that implies from (P1) and (P2). But, does not imply , in general. A basic example of a partial metric space is the pair , where for all .

*Remark 3 (see [16]). *Clearly, a limit of a sequence in a partial metric space need not be unique. Moreover, the function need not be continuous in the sense that and imply . For example, if and for , then for , for each and so, for example, and when .

Proposition 4 (see [17]). *Let and define the partial distance function by
**
For and , respectively. Then, is complete partial metric space where the self-distance for any point is its value itself. The pair is complete partial metric space for which is called the usual partial metric on , and where the self-distance for any point is its absolute value.*

Proposition 5 (see [18]). *If is a partial metric on , then the function defined by
**
is a usual metric on . For example, in , where is the usual partial metric on , we obtain the usual distance in since for any , .*

*Definition 6 (see [15]). *A partial order on is a binary relation on such that (i) (reflexivity), (ii)if and then (antisymmetry), (iii)if and then (transitivity).A partially ordered set (or poset) is a pair such that is a partial order on . For each partial metric space let be the binary relation over such that (to be read, is part of ) if and only if . Then it can be shown that is a poset.

For the partial metric over the nonnegative reals, is the usual ordering. For intervals, if and only if is a subset of .

*Definition 7 (cf. [17–20]). *Let be a partial metric space and a sequence in . Then, we say the following:(a)A sequence converges to a point if and only if .(b)A sequence is a* Cauchy sequence* if there exists (and is finite) .(c)A partial metric space is said to be* complete* if every Cauchy sequence in converges, with respect to the topology , to a point such that . It is easy to see that every closed subset of a complete partial metric space is complete.(d)A mapping is called to be* continuous* at if for every , there exists such that .(e)A sequence in a partial metric space converges to a point , for any such that , there exists so that for any , .

Lemma 8 (see [18]). *Let be a partial metric space. Then,* (i)* is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space ,* (ii)*a partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if .*

In the partial metric space , the limit of the sequence is since one has , where is the usual metric induced by on .

##### 2.1. The Level Sets of Fuzzy Numbers

A* fuzzy number* is a fuzzy set on the real axis, that is, a mapping which satisfies the following four conditions. (i) is normal; that is, there exists an such that . (ii) is fuzzy convex; that is, for all and for all . (iii) is upper semicontinuous. (iv)The set is compact (cf. Zadeh [21]), where denotes the closure of the set in the usual topology of .

We denote the set of all fuzzy numbers on by and call it as* the space of fuzzy numbers*. *-level set * of is defined by
The set is closed, bounded, and nonempty interval for each which is defined by . can be embedded in , since each can be regarded as a fuzzy number defined by

Representation Theorem 1 (see [22]).* Let ** for ** and for each **. Then the following statements hold.* (i)* is a bounded and nondecreasing left continuous function on **.* (ii)* is a bounded and nonincreasing left continuous function on **.* (iii)*The functions ** and ** are right continuous at the point **.* (iv)*.*

*Conversely, if the pair of functions ** and ** satisfies conditions (i)–(iv), then there exists a unique ** such that ** for each **. The fuzzy number ** corresponding to the pair of functions ** and ** is defined by **, **.*

Now we give the definitions of triangular fuzzy numbers with the -level set.

*Definition 9 (triangular fuzzy number, [23, Definition, page 137]). *The membership function of a triangular fuzzy number represented by is interpreted, as follows:
Then, the result holds for each .

Let and . Then the operations addition, scalar multiplication, and product defined on by then .

Let be the set of all closed bounded intervals of real numbers with endpoints and ; that is, . Define the relation on by . Then it can easily be observed that is a metric on (cf. Diamond and Kloeden [24]) and is a complete metric space, (cf. Nanda [25]). Now, we can define the metric on by means of the Hausdorff metric as

Proposition 10 (see [26]). *Let and . Then, the following statements hold.* (i)* is a complete metric space (cf. Puri and Ralescu [27]).* (ii)

*.*(iii)

*.*(iv)

*.*(v)

*.*

*Definition 11 (see [28]). *The following statements hold.(a)A sequence of fuzzy numbers is a function from the set into the set . The fuzzy number denotes the value of the function at and is called as the general term of the sequence.(b)A sequence is called convergent to , if and only if for every there exists an such that for all .(c)A sequence is called bounded if and only if the set of its terms is a bounded set. That is to say that a sequence is said to be bounded if and only if there exist two fuzzy numbers and such that for all . This means that and for all .

The boundedness of the sequence is equivalent to the fact that If the sequence is bounded then the sequences of functions and are uniformly bounded in .

#### 3. Completeness of the Sets of Sequences with Respect to the Partial Metric

Following Kadak and Ozluk [11], we give the classical sets , , , and consisting of the bounded, convergent, null, and -summable sequences of fuzzy level sets with the partial metric , as follows: One can show that , , and are complete metric spaces with the partial metric defined by where and are the elements of the sets , , or . Also, the space is complete metric space with the partial metric defined by where and are the points of .

Theorem 12. *Let denote any of the spaces , , and , and , . Define the partial distance function on by
**
Then, is a complete metric space.*

*Proof. *Since the proof is similar for the spaces and , we prove the theorem only for the space . Let , , and . Then, (i)by using the axiom (P1) in Definition 1, it is trivial that
(ii)By using the axiom (P2) in Definition 1, it follows that
(iii)By using the axiom (P3) in Definition 1, it is clear that
(iv)By using the axiom (P4) in Definition 1 with the inequalities and , we have
Therefore, one can conclude that is a partial metric space on . It remains to prove the completeness of the space . Let be any Cauchy sequence on , where . Then, for any , there exists for all such that
A fortiori, for every fixed and for
Hence for every fixed , by using the completeness of in Theorem 3.1 [11], we say the sequence is a Cauchy sequence and is uniformly convergent. Now, we suppose that and . We must show that
The constant for all , taking the limit for in (22), we obtain
for all . Since , there exists a number such that for all . Thus, (24) gives together with the triangle inequality of partial metric for that
It is clear that (25) holds for every whose right-hand side does not involve . This leads us to the consequence that is bounded sequence of fuzzy numbers hence . Also, from (24) we obtain for that
This shows that (23) holds and . Since is an arbitrary Cauchy sequence, is complete.

Theorem 13. *Define the distance functions , , and by
**
where , are the element of the spaces , , or , respectively. Then, , , and are complete metric spaces.*

*Proof. *Since the proof is similar for the spaces and , we prove the theorem only for the space . One can easily establish that defines a metric on . Let be any Cauchy sequence on . Then for every , there exists a positive integer for all , such that
where and . We obtain for each fixed from (28) that
for all , which leads us to the fact that the sequence is a Cauchy sequence and is convergent. Now, we suppose that as . We have from (29) for each and , that
Take any . Let firstly and nextly in (30) to obtain . Finally, by using Minkowski's inequality for each
which implies that . Since for all , it follows that as . Since is an arbitrary Cauchy sequence, the space is complete. This step concludes the proof.

#### 4. The Duals of the Sets of Sequence with the Partial Metric

The idea of dual sequence space, which plays an important role in the representation of linear functionals and the characterization of matrix transformations between sequence spaces, was introduced by Köthe and Toeplitz [29], whose main results concerned -duals. An account of the duals of sequence spaces can be found in Köthe [30]. One can also find about different types of duals of sequence spaces in Maddox [31].

In this section, we focus on the alpha-, beta- and gamma-duals of the classical sets of sequences of fuzzy numbers with partial metric. For the sets , , and of sequences defined by is called the multiplier sets of and for all . One can easily observe for a sequence set of fuzzy level sets that the inclusions hold. The alpha-, beta- and gamma-duals , , and of a set are, respectively, defined by where the coordinatewise product of the sequences and of level sets for all . Then is called -dual of or the set of all factor sequences of are in . Firstly, we give a remark concerning with the convergence factor sequences of fuzzy level sets with partial metric.

*Remark 14. *Let . Then the following statements are valid.(a) is a set of sequence and (“” stands for “is a linear subset of ”) where
(b)If then .(c).(d) and .

*Proof. *Since the proof is trivial for conditions (b) and (c), we prove only (a) and (d). Let and .(a)Let . Then we get ; and . Since is arbitrary, . For any and we have
and we get . Therefore, is a linear subset of .(d)Using (a) we need only show . Suppose that and be given with geometric division by if and otherwise. By taking into account the set from the case (a), then there exists an integer for all such that . Thus, we have
Further, implies that . The rest is an immediate consequence of this part; we omit the detail.

Theorem 15. *The following statements hold. *(a)*.*(b)*.*

*Proof. *(a) Obviously by Remark 14(b). Then we must show that and . Now, consider and are given. Then
which implies that . So the condition holds.

Conversely, for a given we prove the existence of an with . According to we may take an index sequence which is a strictly increasing real valued sequence with and . If we define by , where the real signum function defined by
for all , thus, we get
for all . Therefore and thus . Hence .

(b) From the condition (c) of Remark 14 we have since . Now we assume the existence of a . Since is an unbounded sequence there exists a subsequence of such that for all . The sequence is defined by if and otherwise. Then . However
Hence , which contradicts our assumption and . This step completes the proof.

Further to the statements in Remark 14 we make the following remarks which are immediate consequences of the definition of the -duals .

*Remark 16. *Let . Then the following statements are valid.(a); in particular, is a set of sequence.(b)If then .(c)If is an index set, if are sets of sequences and if , then
where the notation “” stands for the span of linear subset in .(d)If .

*Proof. *Condition (b) is obviously true, and (a) follows from . We only show conditions (c) and (d) taking . Other parts can be obtained in a similar way.(c)Now, as an immediate consequence that the following conditions
hold by (b). On the other hand, if , that is , then for all and therefore .(d)We prove . Let ; then, for all ; thus, and by (a).

In general as we get from Theorem 15(a) in the case of and . We have . This remark gives rise to the following definition.

*Definition 17 (-space, Köthe space). *Let , and let be a set of sequence. is called -space if . Further, an -space is also called a Köthe space or perfect sequence space.

From Remarks 4.3(d) and (b) we obtain immediately the following remark.

*Remark 18. *If is a set of sequence over real field and , then is a -space; that is, .

Now we look for sufficient conditions for . This gives rise to the notion of solidity.

*Definition 19 (Solidness). *Let be a set of sequence over the field . Then is called solid if

Theorem 20. *Consider is any set of sequence over the field ; then, the following statements hold.*(a)*If is a Köthe space, then is solid.*(b)*If is solid, then .*(c)*If is a Köthe space, then is a -space.*

*Proof. *Let be a set of sequence over the field .(a)If is a Köthe space and , then if and only if the condition holds for all . Besides this we obtain for and and the statement
holds for each . Therefore . Hence and is solid over the real field.(b)Consider is solid. To show , it suffices to verify as we have Remark 16 (a). So, let ; that is,
By taking into account solidness of , for , where and the condition holds; there exists a sequence for all . Therefore by combining this with the inclusion (46) we deduce that the condition
holds and . Hence and .(c)This is an obvious consequence of Remark 18 and conditions (a), (b) in Theorem 20.

Theorem 21. *The following statements hold.*(a)*The sets , , , , and of sequences are solid.*(b)*The sets and of sequences are not solid; therefore, none of them is a Köthe space.*(c)*For each , then(i) and ,(ii) and .*(d)

*If and , then and . In particular , and each of is not a -space.*

*Proof. *Given specified sets are solid in (a) and (b) is an immediate consequence of their definition. Additionally, the parts (i) and (ii) of (c) can be obtained Theorem 15 and Remark 14(d). Since and are solid, we know that . So the statements in (d) obtain from Remark 16(b).

Next, we determine the -duals of the spaces , , , and . We will find that none of these sets is solid; in particular, none of them is a Köthe space.

Theorem 22. *The following statements hold.*(a)*.*(b)*, , , .*(c)*, , , .*

In particular the sets , , , and of sequences are -spaces, but they are not Köthe spaces. Moreover, the sets and of sequences are -spaces, whereas both and are not -spaces. None of the spaces , , , and is solid.

*Proof. *We prove the cases for the spaces , and the proofs of all other cases are quite similar.(a)Let and . Then,
Therefore, which gives that . Conversely, suppose that . Then we can construct an index sequence with and . Define by
Then . According to the choice of the inequalities
hold. Thus , which implies . This contradicts that . Therefore . As well if we take the sequence by