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Connecting Fuzzifying Topologies and Generalized Ideals by Means of Fuzzy Preorders
The present paper investigates the relations between fuzzifying topologies and generalized ideals of fuzzy subsets, as well as constructing generalized ideals and fuzzifying topologies by means of fuzzy preorders. Furthermore, a construction of generalized ideals from preideals, and vice versa, is obtained. As a particular consequence of the results in this paper, a construction of fuzzifying topology generated by generalized ideals of fuzzy subsets via a given -topology is given. The notion of -generalized ideal is introduced and hence every -generalized ideal is shown to be a fuzzifying topology induced by some fuzzy preorder.
The concepts of preideals and generalized ideals were introduced by Ramadan et al. , as a consistent approach to the ideas of fuzzy mathematics. The authors investigated the interrelations between these two concepts. This paper is devoted to finding fuzzifying topologies derived from preideals, and vise versa, by means of fuzzy preorders and residual implications. The contents of this paper are arranged as follows. In Section 2, we recall some basic notions of ideals, preideals, and generalized ideals. The notion of saturation of preideals is introduced, hence, a one-to-one correspondence between the set of saturated preideals and the set of generalized ideals on the same set is obtained. In fact, some studies were done on the correspondence of such two sets, see [2, 3], for example, based on the duality between fuzzy filters and fuzzy ideals. In our paper, we prove such correspondence independent of the duality principle. Also, given a Boolean lattice, we construct a generalized ideal, in terms of fuzzy preorders. Section 3 is concerned with the relationship between the fuzzy preorders and each of -topologies , fuzzifying topologies , and -fuzzy topologies , respectively. Thus, an interesting relation between a special type of -topological spaces, called fuzzy neighborhood spaces, introduced by Lowen , and fuzzifying topological spaces is established. In Section 4, we introduce the notion of -generalized ideal, therefore, a characterization of this special kind of ideals is shown to be a fuzzifying topology generated by some fuzzy preorder. Also, we study the relations between preideals and generalized topological structures, for example, -topologies, fuzzifying topologies, and -fuzzy topologies. Also, -topologies are constructed from preideals via given -topologies.
2. Generalized Ideal Structures
We recall some basic definitions. We should let denote a nonempty set and the closed unit interval , respectively, and we let .
We denote the characteristic function of a subset also by . If , we define . Let be the collection of all subsets of . The fuzzy set which assigns to each element in the value , is also denoted by . In this paper, the concepts of triangular norm and residual implication are applied.
A triangular norm, a -norm in short, on the unit interval is a binary operator which is symmetric, associative, order-preserving on each place and has 1 as the unit element.
A fuzzy relation on a nonempty set is a map . A fuzzy relation is called(1)reflexive if for each ;(2)-transitive if .
A reflexive and -transitive fuzzy relation is called -fuzzy preorder. If is assumed to be the t-norm , we will drop the -.
This section presents a review of some fundamental notions of ideals, preideals, and generalized ideals. We refer to Ramadan et al.  for details.
Definition 2.1. A nonempty subset of is an ideal of subsets of , simply, an ideal on , if it satisfies the following conditions:(D1)(D2)(D3)
Definition 2.2. A nonempty subset of is a preideal of subsets of , simply, a preideal on , if it satisfies the following conditions:(P1);(P2);(P3).
A preidael is called prime if it satisfies the condition;
A preideal is called a -preideal if the following hold.
If is a countable subcollection of , then .
If is a preideal on , we define the characteristic, denoted by , of by
If is a preideal and , we say that is saturated if for every , , equivalently, is saturated if it satisfies
We provide a useful characterization of a saturated preideal in the following lemma.
Lemma 2.3. Let be a saturated preideal on , . Then
Proof. Let . Since for all , then, by (P3), for all .
Conversely, let such that for all in . For every and , we consider the fuzzy subsets . Since is finite, then, by (P2), . Also, it is easy to see that, for all , , hence . Consequently, , since is saturated, so we are done.
It should be noticed that the required condition of Lemma 2.3 holds, trivially, for all and , therefore, we provide , so that the above condition is significant.
Also, the characteristic of a preideal , on a subset of may be defined and denoted by . The following result is useful for our study.
Lemma 2.4. Let on . Then, for every , hence, .
Proof. The statement holds when . Suppose , and denote and . Let satisfy . Then , that is, . Conversely, given , let , hence . This implies, by (P3), , so . Thus, by definition of , we get , this shows that the equality holds.
Definition 2.5. A nonzero function is called a generalized ideal of fuzzy subsets of , simply, a generalized ideal on , if it satisfies the following conditions:
(G2) , for each ;
(G3) if then , for each ;
The conditions (G2) and (G3) are equivalent to:
(G*) for each . Moreover, Since a nonzero function, then by (G3), .
A generalized ideal , is called -generalized ideal if and is called prime if
A generalized ideal on a set can be derived by fuzzy relations in different ways, as we see in the following.
Proposition 2.6. Let be a reflexive fuzzy relation on a Boolean lattice with . Define , for all as follows: Then are generalized ideal on .
Now, we will construct different types of preideals derived from a generalized ideals.
Proposition 2.7. Let be a generalized ideal on . Define by Then both are saturated preideals on .
Proof. First, we show that a preideal on :
(P1) , otherwise for all , that is, , which is a contradiction, since is a nonzero function;
(P2) let . Then for all : hence .
(P3) Let such that and. Then for all , Let be a family of fuzzy sets in . Then for all , therefore, , hence is saturated.
Second, we prove that is saturated preideal on :
(P1) , otherwise , for all ;
(P2), (P3) direct.
Let be a family of fuzzy sets in . Then Hence is saturated preideal.
Proposition 2.8. Let be a prime generalized ideal on . Define by Then are prime saturated preideals on . Moreover .
Proof. First, we show that . Let such that for all . If , we assume that , hence .
Then, by (G3), , that implies .
Conversely, let such that , for all . Then , for all this implies , thus .
Second, we show that is a saturated prime preideal.
(P1) Suppose that . Let . Then, for all , which is a contradiction, so .
(P2) Let . Then for all , , since is prime. This implies .
(P3) Let such that and . Then , by (G3), hence .
Now, we will show that is saturated. Suppose for all . Then for all , , hence , that is, is saturated.
Finally, we show that is prime. Let such that . Then for all , hence, or , that implies or .
Proposition 2.9. Let be a preideal on . Define by Then is a generalized ideal on .
Proof. (G1) , since implies .
(G2) Let and such that .
Then, by (P2), . Let . Then, by (P3), we get , and . Thus , hence , so .
Now we come to the main result of this section.
Theorem 2.10. There is a one-to-one correspondence between the set saturated preideals on a nonempty set and the set of generalized ideals on .
Proof. Let be a generalized ideal on . Denote the preideal generated by , defined as in Proposition 2.7, by , and denote the generalized ideal, generated by , as in Proposition 2.9, by , hence for all Conversely, let be a saturated preideal on . Denote the generalized ideal generated by , as in Proposition 2.9, by , and denote the preideal generated by , as in (2.11) in Proposition 2.7 by . Then, This completes the proof.
Definition 2.11. A nonzero function is called a fuzzy -ideal on if it satisfies the following conditions:
(d2) , for ;
(d3) if , then .
Clearly, a generalized ideal on a set can be regarded as a fuzzy -ideal whose domain is . Precisely, for each ; otherwise .
Remark 2.12 (Fang and Chen ). Defined a generalized fuzzy preorder , corresponding to a map as follows: hence, for a fuzzy -ideal on , there corresponds a fuzzy preorder defined by Especially, for a generalized ideal on , the corresponding fuzzy preorder of is given by the following: For every ,
3. Connecting Fuzzy Preorders and -fuzzy Topologies
In this section, we recall some basic results about the connection between fuzzy preorders and each of -topologies, fuzzifying topologies, and -fuzzy topologies. Also, the relations among these different types of topologies are investigated.
An -topology on a set is a crisp subset of which is closed with respect to finite meets and arbitrary joins and which contains all the constant functions from to . If the constants are only, we will call it a Chang -topology.
Another more consistent approach to the fuzziness has been developed. According to Shostak , an -fuzzy topology on a set is a map , satisfying the following conditions:(1) for all constant functions ;(2), for all ;(3), .
By the conditions () and (), we get for and , . An -fuzzy topology on a set is called trivial, if , for all , and is called homogeneous if the following condition holds: for and , .
If a map satisfies the similar conditions of -fuzzy topology on a set X, then is called a fuzzifying topology Ying . Furthermore, if a fuzzifying topology satisfies the following condition:
() for all , then is called a saturated fuzzifying topology.
For a given left-continuous -norm , the corresponding residual implication is given by .
Many properties of residual implications are found in litrature, but we will recall some of them, which will be used in this paper:(I1) if and only if ;(I2);(I3);(I4), hence, if ;(I5), hence if ;(I6);(I7);(I8);(I9);(I10);(I11);(I12).
Clearly, by (I1) and (I3), the residual implication with respect to a -norm is a fuzzy preorder on and called thecanonical fuzzy preorder on .
The relationships between fuzzy preorders and -topologies, fuzzifying topologies, and -fuzzy topologies, respectively, have been investigated by many authors. For example, we recall the main results of Lai and Zhang , Fang and Chen , and Fang .
Theorem 3.1 (Lai and Zhang ). For a fuzzy preorder on a set , the family , is an -Alexandrov topology induced by , that is, satisfies the folowing properties: for all and ,
(a) every constant fuzzy set belongs to ;
Moreover, satisfies the equality Conversely, for a given -Alexandrov topology a set , there is a unique -fuzzy preorder , given by , such that .
Thus, there is a one-to-one correspondence between the set of -Alexandrov topologies on a set and -fuzzy preorders.
Theorem 3.2 (Fang and Chen ). Every fuzzy preorder on a set corresponds a saturated fuzzifying topology on , given by for all , conversely, every fuzzifying topology on a set corresponds a fuzzy preorder , given by: for all , Thus, there is a one-to-one correspondence between the set of saturated fuzzifying topologies on a set and the set of fuzzy preorders on .
By the above two theorems, one can dedeuce, at once, the following corollary.
Corollary 3.3. The set of saturated fuzzifying toplogies on a set and the set of -Alexandrov topologies on are in one-to-one correspondence.
A special type of -topological spaces, called fuzzy neighborhood spaces, was introduced by Lowen . The interrelations between these spaces and the fuzzifying toplogical spaces are interesting to study. Although a study of fuzzifying topological spaces and their relation with fuzzy neighborhood spaces has been investigated, sestematically, see, for example, , we give, in this section, alternative methods. We recall the folowing definition.
Definition 3.4 (Wuyts et al. ). An -topological space is called a fuzzy neighborhood space if whenever , for any also .
Equivalently, if whenever , for any , also .
Proposition 3.5. Let be a fuzzifying topology on a set . Then corresponds an -topology on , defined by , and is a fuzzy neighborhood space.
Conversely, let be an -topology on . Then corresponds a fuzzifying topology , on , defined by: for all , . Moreover,
and if is a fuzzy neighborhood space, then .
Proof. That is, and are -topology and fuzzifying topology on , respectively, and are easy to be verified. To prove that is a fuzzy neighborhood space, we suppose and , then
Therefore, , that is, , so we are done.
To prove (), let be a fuzzifying topology on then for all ,
To prove (), suppose that is a fuzzy neighborhood space. Then
Therefore, as a direct result of Proposition 3.5, we have the following.
Theorem 3.6. There is a one-to-one correspondence between the set of fuzzifying topologies and the set of fuzzy neighborhood structures on the same set.
Definition 3.7. Given , an -fuzzy topological space, and , then is an -topology on , called the -level -topology of .
Consequently, , by which we will give an interrelation between -topologies and -fuzzy topologies.
Proposition 3.8. Let be an -topology on . Then for all , is a homogeneous -fuzzy topology on .
Proof. That is, an -fuzzy topology is easy to be verified. Now, let , then we get Since is an -fuzzy topology, then , so the equality holds.
Theorem 3.9. There is a one-to-one correspondence between the set of -topologies on a set and the set of homogeneous -fuzzy topologies on .
Proof. Let be an -topology on , then is a homogeneous -fuzzy topology on , by Proposition 3.8, and
Conversely, if is a homogeneous -fuzzy topologies on , then is an -topology and So, we are done.
4. Connecting -fuzzy Topologies and Preideals
In this section, we associate for each a fuzzifying topology, a generalized ideal. Also, we construct a preideal corresponding to an -topology, by means of the residuated implication. Conversely, an -topology, a fuzzifying topology, and -fuzzy topology are derived by old ones via preideals. We made use of the results of Section 2, beside the useful following lemma.
Lemma 4.1. Let . Define by for all Then are fuzzy preorders on , especially, if , then for all ,
Proposition 4.2. Let be an -topology on a set , and be a preideal on . Define , by if and only if . Then is Chang -topology on .
However will be called the -fuzzy preideal topology. As a special case.
Example 4.3. If , and , then . Consequently, has the following property: for all such that , then .
Proposition 4.4. Let be a preideal on a set . Define by for all , . Then is a Chang -fuzzy topology on .
Proof. Clearly, , by (I1). Let . Then , by (P2) and (I12). Let . Then , by (I5) and (P3), so we are done.
The -fuzzy topology , given in the above proposition, is called the -fuzzy topology associated with a preideal and denoted by . It is easy to see that satisfy the following simple properties:(1)for all , ;(2)for all and such that , then ;(3)restricting the range of on , a Chang -topology, associated with a preideal , is obtained, namely,
Proposition 4.5. Let be an -topology on a Boolean lattice with 0. Let , defined as . Then is a saturated preideal on and , where is the generalized ideal corresponding to .
Proof. Let be an -topology on . Then is a fuzzy preorder, by Theorem 3.1, hence is a generalized ideal, by (2.9) in Proposition 2.6. Thus, is a saturated preideal by (2.11) in Proposition 2.7. Secondly, we prove the equality of the two forms of as follows: which completes the proof.
Proposition 4.6. Let be a preideal. Define by Then is an -Alexandrov topology on , containing .
Proposition 4.7. Let be a nontrivial -fuzzy topology on a set . Define , then is a preideal on .
Proposition 4.8. Let be a preideal on a set . Define by . Then is an -topology if is -preideal.
Proof. Since , then contains all the constants fuzzy subsets on . Secondly, if then there exist and such that and , then . By (P3) and (P2), we get , hence .
Let , then for all there exists , such that . Therefore, we get , since is a -preideal.
Proposition 4.9. Let be a fuzzifying topology on a Boolean lattice. Define by for all , . Then is a generalized ideal on .
Conversely, let be a generalized ideal on , then , defined by for all , is a fuzzifying topology on .
Proposition 4.10. Let be a -generalized ideal on a set . Define by for all , .
Then, is a fuzzy preorder and for all , .
Proof. That is a fuzzy preorder follows from Remark 2.12, hence according to Theorem 3.2, for all ,
Secondly, we have to show that for any , .
Let such that . Then for any , there exists such that , and . Let , then and , since is -generelized ideal. Obviously, . So we would get , by (G3). Thus , which completes the proof.
Example 4.11. () Let and then
Then is a preideal on , and
is a fuzzy preorder, hence
Also, another fuzzy preorder may be dedeuced by , Lemma 4.1: Then
() Let , and , then , by applying Proposition 4.5.
The author is grateful to Professor Etienne Kerre for his warm encouragement.
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