Degrees of (L, M)-fuzzy bornologies

Vildan Çetkin

doi:10.1515/math-2024-0110

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Degrees of (L, M)-fuzzy bornologies

Vildan Çetkin

Published/Copyright: December 19, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

This article is devoted to present the degree to which a mapping defined from L X to M , which is an ( L , M ) -fuzzy bornology in the sense of Liang et al. In this respect, the degree to which L -subset is bounded according to the given bornological structure is described. In addition, the notions of boundedness degree and the boundedness preserving degree of a mapping defined between bornological spaces are introduced. The characterizations of the provided notions are studied.

Keywords: fuzzy set; bounded L-set; bornological space; bounded mapping

MSC 2010: 03E72; 03G10; 54A05; 54C10

1 Introduction

Bounded sets described in metric spaces play an important role in some applications, but in general topological spaces, the notion of a “bounded set” makes no sense by the absence of the distance function. Hence, to identify bounded sets independently from the distance function, a structure named bornology (or so called abstract boundedness) has been constructed by Hu [1]. Hence, the conception of boundedness has been applied to the case of a general topological space. According to this definition a bornology is a collection of sets which satisfies some certain conditions: closed for finite unions, closed hereditary and contains all singletons. The sets which belong to a bornology are called as the bounded sets of the space. The families CL ( X ) , F ( X ) , and K ( X ) of all nonempty closed, all nonempty finite and all nonempty compact subsets of a Hausdorff topological space X , the family of all (totally) bounded subsets of a metric or uniform space are examples of boundedness. At present, the theory of bornological spaces is developed in various directions. Most of the research involving bornologies is done in the context of topological linear spaces [2] and in topological algebras, i.e., in case when the underlying set, in addition to topology, is endowed with a certain algebraic structure. Maio and Kočinac [3] studied the notion of boundedness in a topological space and demonstrated the importance of this notion in selection principles theory. Caserta et al. [4] investigated some properties of the function spaces endowed with bornologies in the view of selection principles.

With the development of fuzzy mathematics, many mathematical structures have been generalized to the fuzzy case, such as fuzzy control, fuzzy topology, fuzzy algebra, and so on. So in a natural way, the notion of bornology has been combined with fuzzy set theory. In 2011, Abel and Sostak [5] first generalized the notion of axiomatic bornology to the fuzzy case, which is called an L -bornology. Hence, they described the concept of classical boundedness for L -fuzzy sets. In the setting of L-bornology, Paseka et al. [6] concerned some categorical properties and showed that the category of (strong) L -bornology is topological universe. Zhang and Zhang [7] proposed the concept of I -bornological vector spaces and discussed two methods on constructing new I -bornological vector spaces. In 2013, Šostak and Uljane [8] discussed two alternative approaches how the concepts of bornology and boundedness can be extended to the case of fuzzy sets and many-valued structures. In this work, they gave birth to the definition of L -valued bornology, which is a mapping from 2 X (the powerset of X) to L satisfying L -valued analogues of the axioms of a bornology. They also proposed L -valued bornologies induced by fuzzy metrics.

In 2017, Šostak and Uljane [9] extended L -valued bornology to M -bornology on a L -valued set by means of L -valued equality, which is called L M -bornology, for short. In this article, they initiated the study of the category of L M -bornological spaces and appropriately defined bounded mappings of such spaces. In 2023, Liang et al. [10] proposed a new kind of M -valued L -fuzzy bornology, namely, ( L , M ) -fuzzy bornology. In this article, they studied the fundamental properties of the ( L , M ) -fuzzy bornology and also they defined ( L , M ) -fuzzy bornological vector spaces. In addition, they observed the constructed structures in the categorical aspect. Parameterized extension of the L M -bornology was studied by Çetkin [11,12]. Further, the theory of bornological spaces is developed in various directions by many authors [13–16].

Nowadays, many topological properties have been endowed with degrees in ( L , M ) -fuzzy topological spaces. In particular, the degrees of axioms related to ( L , M ) -fuzzy topology and the degrees of continuous mappings between ( L , M ) -fuzzy topological spaces were presented by Kubiak and Šostak [17,18]. In 2014, Liang and Shi [19] presented the degrees to which a mapping is continuous, open or closed in ( L , M ) -fuzzy topological spaces in a different way. Zhong and Shi [20] discussed the characterizations of ( L , M ) -fuzzy topology degrees. Zhong et al. [21] introduced the degree to which a mapping defined from L X to M is an ( L , M ) -fuzzy convexity.

The former studies inspired us, and we decided to make contribution to the gradation of ( L , M ) -fuzzy topological structures. In this respect, we found it reasonable to study the degree approach of the ( L , M ) -fuzzy bornology. Hence, each mapping defined from L X to M can be regard as bornology to some degree. The structure of this article is organized as follows. In Section 2, we recall some necessary notions and notations for the sequel. In Section 3, we give the degree to which a mapping ℬ : L X → M is an ( L , M ) -fuzzy bornology in the sense of Liang et al. [10]. We also present the degree to which an L -subset is an M -bounded set with respect to the corresponding ℬ . Afterwards, when ( X , ℬ 1 ) and ( Y , ℬ 2 ) are ( L , M ) -fuzzy bornological spaces to some extent, the degrees to which a mapping φ defined between the given spaces, is a bounded mapping or a boundedness preserving mapping are provided. We investigate the relations between them and also observe the characterizations of the presented concepts.

2 Preliminaries

Throughout this article, unless otherwise stated, L = ( L , ≤ , ∧ , ∨ ) denotes a complete lattice and M = ( M , ≤ , ∧ , ∨ ) denotes a completely distributive lattice. The bottom and the top elements of M are denoted by 0 M and 1 M , respectively. Similarly, the bottom and the top elements of L are denoted by 0 L and 1 L , respectively. X refers to a non-empty crisp set. By L X , we denote the collection of all L -subsets on X .

For each x ∈ X and λ ∈ L , the L -subset x λ , defined by x λ ( y ) = λ if y = x , and x λ ( y ) = 0 L , if y ≠ x , is called a fuzzy point. An element a ∈ M is called a prime element if a ≥ b ∧ c implies a ≥ b or a ≥ c . The set of all non-unit prime elements is denoted by p ( L ) . An element a ∈ M is called a co-prime element if a ≤ b ∨ c implies a ≤ b or a ≤ c . The set of all non-zero co-prime elements is denoted by c ( L ) . It is obvious that the set of all fuzzy points c ( L X ) is exactly the set of all co-prime elements of L X . For more details about the lattices, we refer [22,23].

Let φ : X → Y be a classical function. The forward L -powerset operator φ → : L X → L Y and the backward L -powerset operator φ ← : L Y → L X induced by φ are defined by φ → ( A ) ( y ) = ⋁ y = φ ( x ) A ( x ) for all A ∈ L X and y ∈ Y , and φ ← ( B ) ( x ) = B ( φ ( x ) ) , for all B ∈ L Y and x ∈ X .

The binary operation ↦ which is called residual implication in the completely distributive lattice M is given by a ↦ b = ∨ { c ∈ M ∣ a ∧ c ≤ b } .

Further the operator ↔ is given by a ↔ b = ( a ↦ b ) ∧ ( b ↦ a ) .

Lemma 2.1

[24] Let M = ( M , ∨ , ∧ ) be a completely distributive lattice and ↦ be the implication operation corresponding to ∧ . Then for all a , b , c ∈ M and { a i } i ∈ I , { b i } i ∈ I ⊆ M , the following conditions are satisfied.

1 M ↦ a = a .
( a ↦ b ) ≥ c ⇔ a ∧ c ≤ b .
a ↦ b = 1 M ⇔ a ≤ b .
a ↦ ( ⋀ i ∈ I b i ) = ⋀ i ∈ I ( a ↦ b i ) , hence a ↦ b ≤ a ↦ c whenever b ≤ c .
( ∨ i ∈ I a i ) ↦ b = ⋀ i ∈ I ( a i ↦ b ) , hence a ↦ c ≥ b ↦ c whenever a ≤ b .
( a ↦ c ) ∧ ( c ↦ b ) ≤ a ↦ b .

The binary relation ◃ in M is defined as follows: for a , b ∈ M , a ◃ b if and only if for every subset D ⊆ M , the relation ⋀ D ≤ b always implies the existence of d ∈ D with d ≤ a . Let α ( b ) = { a ∈ M ∣ a ◃ b } . We have b = ⋀ α ( b ) for each b ∈ M , and the mapping α : M → 2 M is ⋀ − ⋃ and order-reversing (see [25,26]).

In 2011, Abel and Sostak [5] first generalized the notion of axiomatic bornology to the fuzzy case, which is called L-bornology.

Definition 2.1

[5] A subcollection ℬ ⊆ L X is called an L -bornology on X if the following axioms are satisfied.

(LB1) ∨ B ∈ ℬ B ( x ) = 1 M , for all x ∈ X .

(LB2) For all B ∈ ℬ with A ≤ B implies A ∈ ℬ .

(LB3) If A , B ∈ ℬ , then A ∨ B ∈ ℬ .

Then the pair ( X , ℬ ) is called an L -bornological space. Given L -bornological spaces ( X , ℬ X ) and ( Y , ℬ Y ) , a mapping φ : ( X , ℬ X ) → ( Y , ℬ Y ) is called L -bounded if φ → ( B ) ∈ ℬ Y for all B ∈ ℬ X .

In [5], a stronger version of the axiom (LB1) was considered:

(LB1*) ∀ x ∈ X , χ { x } ∈ ℬ .

The corresponding structure ℬ is called a strong (or strict) L -bornology on X .

In 2017, Sostak and Uljane [9] extended L -bornology to M -subset of L X , which is called L M -bornology as follows:

Definition 2.2

[9] A mapping ℬ : L X → M which satisfies the following three axioms is called an L M -bornology on ( X , E ) , where E is an L -valued equality on X .

(1LMB) ∀ a ∈ M o , ∃ U ⊆ L X s.t. ∨ U = 1 L X and ∀ U ∈ U , ℬ ( U ) ≥ a , where M o = { a ∈ M ∣ a ◃ 1 M } .

(2LMB) A ≤ B implies ℬ ( A ) ≥ ℬ ( B ) for all A , B ∈ L X .

(3LMB) ℬ ( A ∨ B ) ≥ ℬ ( A ) ∧ ℬ ( B ) for all A , B ∈ L X .

(4LMB) ℬ ( e ( A ) ) = ℬ ( A ) .

Then the triple ( X , E , ℬ ) is called an L M -bornological space.

In 2023, Liang et al. [10] introduced a new kind of M -valued L -fuzzy bornology which is called ( L , M ) -fuzzy bornology as follows. According to this axiomatic approach, ( L , M ) -fuzzy bornology is the most general one in the sense that it can contain L -bornology, strong L -bornology [5], and M -fuzzifying bornology [8] as special cases.

Definition 2.3

[10] A mapping ℬ : L X → M which satisfies the following three axioms is called an ( L , M ) -fuzzy bornology on X .

(LMB1) ℬ ( x 1 L ) = 1 M .

(LMB2) For each A , B ∈ L X with A ≤ B implies ℬ ( A ) ≥ ℬ ( B ) .

(LMB3) ℬ ( A ∨ B ) ≥ ℬ ( A ) ∧ ℬ ( B ) , for any A , B ∈ L X .

Then the pair ( X , ℬ ) is called an ( L , M ) -fuzzy bornological space. The value ℬ ( A ) can be interpreted as the degree of boundedness of the L -fuzzy set A .

Remark 2.4

[10]

It is clear from (LMB1) and (LMB2), ℬ ( x λ ) = 1 M , for each x λ ∈ c ( L X ) .
( L , M ) -fuzzy bornology can degenerate to L -bornology (strong L -bornology), M -fuzzifying bornology and classical bornology by restricting M = { 0 , 1 } , L = { 0 , 1 } and L = M = { 0 , 1 } , respectively.

Proposition 2.1

[10] If ( X , ℬ ) is an ( L , M ) -fuzzy bornological space, then the followings are satisfied.

The sets ℬ [ a ] = { A ∈ L X ∣ ℬ ( A ) ≥ a } , for all a ∈ M is an L-bornology on X .
The sets ℬ [ a ] = { A ∈ L X ∣ a ∉ α ( ℬ ( A ) ) } , for all a ∈ p ( M ) is an L -bornology on X .

Definition 2.5

Let ( X , ℬ X ) and ( Y , ℬ Y ) be two ( L , M ) -fuzzy bornological spaces. Then the mapping φ : ( X , ℬ X ) → ( Y , ℬ Y ) is called

an ( L , M ) -fuzzy bounded map if ℬ X ( A ) ≤ ℬ Y ( φ → ( A ) ) , for all A ∈ L X ;
an ( L , M ) -fuzzy boundedness preserving map if ℬ Y ( B ) ≤ ℬ X ( φ ← ( B ) ) , for all B ∈ L Y .

3 Degrees of ( L , M ) -fuzzy bornologies

In this section, we shall consider the following natural question. For any mapping ℬ : L X → M is it an ( L , M ) -fuzzy bornology to some extent? The answer of the previous question gives rise to the notion of the degree to which a mapping ℬ : L X → M is an ( L , M ) -fuzzy bornology.

Definition 3.1

Let ℬ : L X → M be a mapping. Then D ( ℬ ) which is described by

D ( ℬ ) = ⋀ x ∈ X ℬ ( x 1 L ) ∧ ⋀ A , B ∈ L X , A ≤ B ( ℬ ( B ) ↦ ℬ ( A ) ) ∧ ⋀ A , B ∈ L X ( ( ℬ ( A ) ∧ ℬ ( B ) ) ↦ ℬ ( A ∨ B ) )

is said the degree to which ℬ is an ( L , M ) -fuzzy bornology on X .

Remark 3.2

If D ( ℬ ) = 1 M , then for all x ∈ X , ℬ ( x 1 L ) = 1 M , for all A , B ∈ L X with A ≤ B , ℬ ( B ) ≤ ℬ ( A ) and for all A , B ∈ L X , ℬ ( A ∨ B ) ≥ ℬ ( A ) ∧ ℬ ( B ) . It is exactly the definition of ( L , M ) -fuzzy bornology. In addition, ( X , ℬ ) is an ( L , M ) -fuzzy bornological space if and only if D ( ℬ ) = 1 M . However, there exist examples to show that D ( ℬ ) ≠ 1 M .

Example 3.3

Let M = [ 0 , 1 ] and ℬ : L X → M be a mapping which is determined by ℬ ( A ) = 0.5 , for all A ∈ L X . By Definition 3.1, it is seen that the degree D ( ℬ ) = 0.5 ≠ 1 M . This shows that the previous mapping is not an exactly ( L , M ) -fuzzy bornology, but it is bornology to some degree.

Example 3.4

Let X be any non-empty set and L = M = [ 0 , 1 ] . Define a mapping ℬ : L X → M by follows:

ℬ ( A ) = 1 , if A = 0 L X , 0.4 , if A = x λ , λ ∈ ( 0 , 1 ] , x ∈ X , 0.6 , otherwise .

By Definition 3.1, it is seen that D ( ℬ ) = 0.4 ≠ 1 M . This shows that the mapping ℬ defined above is not an ( L , M ) -fuzzy bornology on X exactly, but it is ( L , M ) -fuzzy bornology to some degree with 0.4 .

Example 3.5

Let X = { x , y , z } be a given set and L = M = [ 0 , 1 ] . Define a mapping ℬ : L X → M by follows:

ℬ ( A ) = 0.5 , if A = 0 L X , 0.3 , if 0 L X ≠ A ≤ χ { x , y } , 1 , otherwise ,

where χ { x , y } demotes the characteristic function of the set { x , y } . Then according to Definition 3.1, the value D ( ℬ ) = 0.3 ∧ 1 ∧ 0.3 = 0.3 ≠ 1 M .

Example 3.6

Let X be any infinite set and L = M = [ 0 , 1 ] . Define a mapping ℬ : L X → M by follows:

ℬ ( A ) = 1 , if A = 0 L X or A satisfying { x ∣ A ( x ) ≠ 0 L } is a finite set , 0.2 , otherwise .

Then the value D ( ℬ ) = 1 M , i.e., ( X , ℬ ) is an ( L , M ) -fuzzy bornological space.

Example 3.7

Let X = { x , y } be a given set, L = M = { 0 , a , b , 1 } be the diamond-type lattice and A , B ∈ L X be defined as follows: A ( x ) = B ( y ) = a , A ( y ) = B ( x ) = b . Define a mapping ℬ : L X → M by follows:

ℬ ( C ) = a , if 0 L X ≠ C ≤ A , b , if 0 L X ≠ C ≤ B , 1 , otherwise .

Then the value D ( ℬ ) = 1 M ∧ 0 M ∧ 1 M = 0 M , i.e., ( X , ℬ ) is not an ( L , M ) -fuzzy bornological space.

Lemma 3.1

Let ℬ : L X → M be a mapping. For any a ∈ M , a ≤ D ( ℬ ) if and only if a ≤ ℬ ( x 1 L ) for all x ∈ X , ℬ ( B ) ∧ a ≤ ℬ ( A ) for all A , B ∈ L X with A ≤ B and ( ℬ ( A ) ∧ ℬ ( B ) ) ∧ a ≤ ℬ ( A ∨ B ) for all A , B ∈ L X .

Theorem 3.8

Let ℬ : L X → M be a mapping. Then the following is valid.

D ( ℬ ) = ⋁ { a ∈ M ∣ a ≤ ⋀ x ∈ X ℬ ( x 1 L ) , ℬ ( B ) ∧ a ≤ ℬ ( A ) , ∀ A , B ∈ L X with A ≤ B , ( ℬ ( A ) ∧ ℬ ( B ) ) ∧ a ≤ ℬ ( A ∨ B ) , ∀ A , B ∈ L X } .

Proof

The proof is straightforward by the previous lemma.□

Theorem 3.9

Let ℬ : L X → M be a mapping. Then the following characterization is satisfied.

D ( ℬ ) = ⋁ { a ∈ M ∣ ∀ b ≤ a , ℬ [ b ] is an L -bornology } .

Proof

Let a ∈ M be arbitrary and suppose that a ≤ ℬ ( x 1 L ) , for each x ∈ X , ℬ ( B ) ∧ a ≤ ℬ ( A ) , for each A , B ∈ L X with A ≤ B and ( ℬ ( A ) ∧ ℬ ( B ) ) ∧ a ≤ ℬ ( A ∨ B ) , for each A , B ∈ L X . Then for any b ≤ a , we have for each x ∈ X ℬ ( x 1 L ) ≥ b and for B ∈ ℬ [ b ] and A ∈ L X such that A ≤ B , we have ℬ ( A ) ≥ ℬ ( B ) ∧ a ≥ b ∧ a = b , and also for any A , B ∈ ℬ [ b ] , we have ℬ ( A ∨ B ) ≥ ( ℬ ( A ) ∧ ℬ ( B ) ) ∧ a ≥ b ∧ a = b . These imply that x 1 L ∈ ℬ [ b ] , for each x ∈ X and if B ∈ ℬ [ b ] with A ≤ B , then A ∈ ℬ [ b ] . And also if A , B ∈ ℬ [ b ] , then A ∨ B ∈ ℬ [ b ] . It follows that D ( ℬ ) ≤ ∨ { a ∈ M ∣ ∀ b ≤ a , ℬ [ b ] is an L -bornology } .

Conversely, assume that ∀ b ≤ a , ℬ [ b ] is an L -bornology. Let b = a . Then x 1 L ∈ ℬ [ a ] , which means that ℬ ( x 1 L ) ≥ a , for all x ∈ X . Let A , B ∈ L X with A ≤ B and let b = ℬ ( B ) ∧ a . Then b ≤ a and B ∈ ℬ [ b ] . So A ∈ ℬ [ b ] , i.e. ℬ ( A ) ≥ b = ℬ ( B ) ∧ a . Let b = ( ℬ ( A ) ∧ ℬ ( B ) ) ∧ a . Then b ≤ a and A , B ∈ ℬ [ b ] . Hence A ∨ B ∈ ℬ [ b ] , i.e. ℬ ( A ∨ B ) ≥ b = ( ℬ ( A ) ∧ ℬ ( B ) ) ∧ a . Therefore, by the previous theorem, D ( ℬ ) ≥ ∨ { a ∈ M ∣ ∀ b ≤ a , ℬ [ b ] is an L -bornology } .

This completes the proof.□

Theorem 3.10

Let ℬ : L X → M be a mapping. Then the following characterization is satisfied.

D ( ℬ ) = ⋁ { a ∈ M ∣ ∀ b ∉ α ( a ) , ℬ [ b ] is an L - bornology } .

Proof

Let a ∈ M be arbitrary given such that a ≤ ℬ ( x 1 L ) for each x ∈ X , ℬ ( B ) ∧ a ≤ ℬ ( A ) for each A , B ∈ L X with A ≤ B and ( ℬ ( A 1 ) ∧ ℬ ( A 2 ) ) ∧ a ≤ ℬ ( A 1 ∨ A 2 ) for all A 1 , A 2 ∈ L X . Then for any b ∉ α ( a ) , B ∈ ℬ [ b ] with A ≤ B and for all A 1 , A 2 ∈ ℬ [ b ] , it is obtained that b ∉ α ( a ) ∪ α ( ℬ ( B ) ) and b ∉ α ( a ) ∪ α ( ℬ ( A 1 ) ) ∪ α ( ℬ ( A 2 ) ) . It is seen that α ( a ) ∪ α ( ℬ ( B ) ) = α ( a ∧ ℬ ( B ) ) ⊇ α ( ℬ ( A ) ) and also α ( a ) ∪ α ( ℬ ( A 1 ) ) ∪ α ( ℬ ( A 2 ) ) = α ( a ∧ ℬ ( A 1 ) ∧ ℬ ( A 2 ) ) ⊇ α ( ℬ ( A 1 ∨ A 2 ) ) . Hence b ∉ α ( ℬ ( A ) ) and b ∉ α ( ℬ ( A 1 ∨ A 2 ) ) . These imply that A ∈ ℬ [ b ] and A 1 ∨ A 2 ∈ ℬ [ b ] . Since α ( a ) ⊇ α ( ℬ ( x 1 L ) ) for each x ∈ X , it is known that b ∉ α ( ℬ ( x 1 L ) ) . It implies that x 1 L ∈ ℬ [ b ] for each x ∈ X . By Theorem 3.1, it follows that D ( ℬ ) ≤ ∨ { a ∈ M ∣ ∀ b ∉ α ( a ) , ℬ [ b ] is an L -bornology } .

Conversely, let ℬ [ b ] be an L -bornology for any b ∉ α ( a ) . Let b ∉ α ( a ) be arbitrary chosen, then it is seen that x 1 L ∈ ℬ [ b ] , i.e. b ∉ α ( ℬ ( x 1 L ) ) for each x ∈ X . From the arbitrariness of b , it is obtained that α ( ℬ ( x 1 L ) ) ⊆ α ( a ) which implies that a ≤ ℬ ( x 1 L ) for each x ∈ X . For any A , B ∈ L X such that A ≤ B , choose any b ∉ α ( ℬ ( B ) ∧ a ) . Since α ( ℬ ( B ) ∧ a ) = α ( ℬ ( B ) ) ∪ α ( a ) , it is obtained that b ∉ α ( a ) and b ∉ α ( ℬ ( B ) ) . Hence, B ∈ ℬ [ b ] and since ℬ [ b ] is an L -bornology, it is seen that A ∈ ℬ [ b ] , i.e. b ∉ α ( ℬ ( A ) ) . Form the arbitrariness of b , it is obtained that α ( ℬ ( A ) ) ⊆ α ( ℬ ( B ) ∧ a ) . As a result, ℬ ( B ) ∧ a ≤ ℬ ( A ) is got. For any A 1 , 2 ∈ L X , choose arbitrary b ∉ α ( ℬ ( A 1 ) ∧ ℬ ( A 2 ) ∧ a ) . Since α ( ℬ ( A 1 ) ∧ ℬ ( A 2 ) ∧ a ) = α ( ℬ ( A 1 ) ) ∪ α ( ℬ ( A 2 ) ) ∪ α ( a ) is satisfied, it is seen that b ∉ α ( a ) , b ∉ α ( ℬ ( A 1 ) ) and b ∉ α ( ℬ ( A 2 ) ) . These imply that A 1 , A 2 ∈ ℬ [ b ] . Since ℬ [ b ] is an L -bornology, A 1 ∨ A 2 ∈ ℬ [ b ] , i.e. b ∉ α ( ℬ ( A 1 ∨ A 2 ) ) . Since b is arbitrary, it is satisfied that α ( ℬ ( A 1 ∨ A 1 ) ) ⊆ α ( ℬ ( A 1 ) ∧ ℬ ( A 2 ) ∧ a ) , i.e. ℬ ( A 1 ) ∧ ℬ ( A 2 ) ∧ a ≤ ℬ ( A 1 ∨ A 2 ) . By Theorem 3.1, it follows that D ( ℬ ) ≥ ∨ { a ∈ M ∣ ∀ b ∉ α ( a ) , ℬ [ b ] is an L -bornology } .

This completes the proof.□

Theorem 3.11

Let { ℬ λ ∣ ℬ λ : L X → M } λ ∈ Λ be a family of mappings. Then the following inequality holds

⋀ λ ∈ Λ D ( ℬ λ ) ≤ D ( ⋀ λ ∈ Λ ℬ λ ) .

Proof

Let { ℬ λ ∣ ℬ λ : L X → M } λ ∈ Λ be a family of mappings. Then by Lemma 2.1, the following inequality is satisfied.

□ D ( ⋀ λ ∈ Λ ℬ λ ) = ⋀ x ∈ X ⋀ λ ∈ Λ ℬ λ ( x 1 L ) ∧ ⋀ A , B ∈ L X , A ≤ B ( ⋀ λ ∈ Λ ℬ λ ( B ) ↦ ⋀ λ ∈ Λ ℬ λ ( A ) ) ∧ ⋀ A 1 , A 2 ∈ L X ( ( ⋀ λ ∈ Λ ℬ λ ( A 1 ) ∧ ⋀ λ ∈ Λ ℬ λ ( A 2 ) ) ↦ ⋀ λ ∈ Λ ℬ λ ( A 1 ∨ A 2 ) ) = ⋀ x ∈ X ⋀ λ ∈ Λ ℬ λ ( x 1 L ) ∧ ⋀ A , B ∈ L X , A ≤ B ⋀ λ ∈ Λ ( ⋀ λ ∈ Λ ℬ λ ( B ) ↦ ℬ λ ( A ) ) ∧ ⋀ A 1 , A 2 ∈ L X ⋀ λ ∈ Λ ( ( ⋀ λ ∈ Λ ℬ λ ( A 1 ) ∧ ⋀ λ ∈ Λ ℬ λ ( A 2 ) ) ↦ ℬ λ ( A 1 ∨ A 2 ) ) ≥ ⋀ λ ∈ Λ ⋀ x ∈ X ℬ λ ( x 1 L ) ∧ ⋀ λ ∈ Λ ⋀ A , B ∈ L X , A ≤ B ( ℬ λ ( B ) ↦ ℬ λ ( A ) ) ∧ ⋀ λ ∈ Λ ⋀ A 1 , A 2 ∈ L X ( ( ℬ λ ( A 1 ) ∧ ℬ λ ( A 2 ) ) ↦ ℬ λ ( A 1 ∨ A 2 ) ) = ⋀ λ ∈ Λ D ( ℬ λ ) .

In what follows, we introduce the degree to which L -subset is an M -bounded set with respect to ℬ .

Definition 3.12

Given a mapping ℬ : L X → M and an L -fuzzy set A ∈ L X , we define a mapping Born ℬ : L X → M as follows: for all A ∈ L X ,

Born ℬ ( A ) = D ( ℬ ) ∧ ℬ ( A ) .

Then the value Born ℬ ( A ) is called the degree to which A is an M -bounded set with respect to ℬ .

Remark 3.13

If D ( ℬ ) = 1 M , which means ℬ is an ( L , M ) -fuzzy bornology, then Born ℬ ( A ) = ℬ ( A ) , for any A ∈ L X which can be regarded as a generalization of bounded set degree given in [10].

Theorem 3.14

Given a mapping ℬ : L X → M , the followings are satisfied.

For each A , B ∈ L X with A ≤ B implies Born ℬ ( A ) ≥ Born ℬ ( B ) .
Born ℬ ( A ∨ B ) ≥ Born ℬ ( A ) ∧ Born ℬ ( B ) , for any A , B ∈ L X .

Proof

(1) Let A , B ∈ L X such that A ≤ B be given. From Definition 3.8, it suffices to obtain that D ( ℬ ) ∧ ℬ ( B ) ≤ D ( ℬ ) ∧ ℬ ( A ) . Since D ( ℬ ) ∧ ℬ ( B ) ≤ D ( ℬ ) is clear, it is enough to prove that D ( ℬ ) ∧ ℬ ( B ) ≤ ℬ ( A ) for any A , B ∈ L X such that A ≤ B . From Definition 3.1, D ( ℬ ) ∧ ℬ ( B ) ≤ ( ℬ ( B ) ↦ ℬ ( A ) ) ∧ ℬ ( B ) . By Lemma 2.1, ( ℬ ( B ) ↦ ℬ ( A ) ) ∧ ℬ ( B ) ≤ ℬ ( A ) is seen. Therefore, by the former inequalities, the fact that D ( ℬ ) ∧ ℬ ( B ) ≤ ℬ ( A ) is obtained for any A , B ∈ L X such that A ≤ B . This witnesses the proof.

(2) From Definition 3.8, it is known that Born ℬ ( A ∨ B ) = D ( ℬ ) ∧ ℬ ( A ∨ B ) and Born ℬ ( A ) = D ( ℬ ) ∧ ℬ ( A ) , Born ℬ ( B ) = D ( ℬ ) ∧ ℬ ( B ) . Hence, for the proof, it is suffices to to show that D ( ℬ ) ∧ ( ℬ ( A ) ∧ ℬ ( B ) ) ≤ D ( ℬ ) ∧ ℬ ( A ∨ B ) . Since the inequality D ( ℬ ) ∧ ( ℬ ( A ) ∧ ℬ ( B ) ) ≤ D ( ℬ ) is obvious, it is enough to show that D ( ℬ ) ∧ ( ℬ ( A ) ∧ ℬ ( B ) ) ≤ ℬ ( A ∨ B ) . From Definition 3.1, D ( ℬ ) ∧ ( ℬ ( A ) ∧ ℬ ( B ) ) ≤ ( ( ℬ ( A ) ∧ ℬ ( B ) ) ↦ ℬ ( A ∨ B ) ) ∧ ( ℬ ( A ) ∧ ℬ ( B ) ) is obtained. Hence, by Lemma 2.1, ( ( ℬ ( A ) ∧ ℬ ( B ) ) ↦ ℬ ( A ∨ B ) ) ∧ ( ℬ ( A ) ∧ ℬ ( B ) ) ≤ ℬ ( A ∨ B ) is true. As a result, the desired inequality D ( ℬ ) ∧ ( ℬ ( A ) ∧ ℬ ( B ) ) ≤ ℬ ( A ∨ B ) is obtained. This witnesses the proof.□

According to this result, the mapping Born B : L X → M satisfies the conditions (LMB2) and (LMB3).

Definition 3.15

The mappings ℬ X : L X → M and ℬ Y : L Y → M be given. And let φ : X → Y be a given function. Then

the boundedness degree of φ with respect to ℬ X and ℬ Y is defined by
BD ( φ ) = ⋀ A ∈ L X ( Born ℬ X ( A ) ↦ Born ℬ Y ( φ → ( A ) ) ) ;
the boundedness preserving degree of φ with respect to ℬ X and ℬ Y is defined by
BP ( φ ) = ⋀ B ∈ L Y ( Born ℬ Y ( B ) ↦ Born ℬ X ( φ ← ( B ) ) ) .

Theorem 3.16

Let ℬ X : L X → M and ℬ Y : L Y → M be given two mappings, and the values D ( ℬ X ) and D ( ℬ Y ) denotes the ( L , M ) -fuzzy bornological degrees of ℬ X and ℬ Y , respectively. Then the following characterizations for boundedness preserving degree of φ are valid.

BP ( φ ) = ∨ { a ∈ M ∣ D ( ℬ Y ) ∧ ℬ Y ( B ) ∧ a ≤ D ( ℬ X ) ∧ ℬ X ( φ ← ( B ) ) , ∀ B ∈ L Y } .
BP ( φ ) = ∨ { a ∈ M ∣ ∀ b ≤ D ( ℬ Y ) ∧ a , ∀ B ∈ ( ℬ Y ) [ b ] , b ≤ D ( ℬ X ) , φ ← ( B ) ∈ ( ℬ X ) [ b ] } .

Proof

(1) Let a ∈ M be arbitrary given. Then we have, a ≤ BP ( φ ) if and only if a ∧ Born ℬ Y ( B ) ≤ Born ℬ X ( φ ← ( B ) ) , for any B ∈ L Y if and only if D ( ℬ Y ) ∧ ℬ Y ( B ) ∧ a ≤ D ( B X ) ∧ ℬ X ( φ ← ( B ) ) , for any B ∈ L Y . This completes the proof.

(2) Suppose that D ( ℬ Y ) ∧ ℬ Y ( B ) ∧ a ≤ D ( ℬ X ) ∧ ℬ X ( φ ← ( B ) ) , for each B ∈ L Y . Then for any b ≤ D ( ℬ Y ) ∧ a and B ∈ ( ℬ Y ) [ b ] , i.e. b ≤ ℬ Y ( B ) , it is obtained that b ≤ D ( ℬ X ) ∧ ℬ X ( φ ← ( B ) ) . This implies that b ≤ ℬ X ( φ ← ( B ) ) , i.e. φ ← ( B ) ∈ ( ℬ X ) [ b ] . According to the first result, it is seen that BP ( φ ) ≤ ∨ { a ∈ M ∣ ∀ b ≤ D ( ℬ Y ) ∧ a , ∀ B ∈ ( ℬ Y ) [ b ] , b ≤ D ( ℬ X ) , φ ← ( B ) ∈ ( ℬ X ) [ b ] } .

Conversely, let us take any b ≤ D ( ℬ Y ) ∧ ℬ Y ( B ) ∧ a . Then b ≤ D ( ℬ Y ) ∧ a and b ≤ ℬ Y ( B ) , i.e. B ∈ ( ℬ Y ) [ b ] . Hence, from the hypothesis, it is obtained that b ≤ D ( ℬ X ) and φ ← ( B ) ∈ ( ℬ X ) [ b ] , i.e. b ≤ ℬ X ( φ ← ( B ) ) . By the arbitrariness of b , it is seen that D ( ℬ Y ) ∧ ℬ Y ( B ) ∧ a ≤ D ( ℬ X ) ∧ ℬ X ( φ ← ( B ) ) . According to the first condition, it is obtained that BP ( φ ) ≥ ∨ { a ∈ M ∣ ∀ b ≤ D ( ℬ Y ) ∧ a , ∀ B ∈ ( ℬ Y ) [ b ] , b ≤ D ( ℬ X ) , φ ← ( B ) ∈ ( ℬ X ) [ b ] } .

These implications complete the proof.□

Theorem 3.17

BD ( φ ) = ∨ { a ∈ M ∣ D ( ℬ X ) ∧ ℬ X ( A ) ∧ a ≤ D ( ℬ Y ) ∧ ℬ Y ( φ → ( A ) ) , ∀ A ∈ L X } .
BD ( φ ) = ∨ { a ∈ M ∣ ∀ b ≤ D ( ℬ X ) ∧ a , ∀ A ∈ ( ℬ X ) [ b ] , b ≤ D ( ℬ Y ) , φ → ( A ) ∈ ( ℬ Y ) [ b ] } .

Proof

(1) Let a ∈ M be given arbitrary. Then, a ≤ BD ( φ ) if and only if a ∧ Born ℬ X ( A ) ≤ Born ℬ Y ( φ → ( A ) ) , for each A ∈ L X if and only if a ∧ D ( ℬ X ) ∧ ℬ X ( A ) ≤ D ( ℬ Y ) ∧ ℬ Y ( φ → ( A ) ) , for each A ∈ L X . This completes the proof.

(2) It is similarly proved to that of Theorem 3.6 (2).□

Definition 3.18

Let mappings ℬ X : L X → M and ℬ Y : L Y → M be given. If the function φ : X → Y is bijective, then the isomorphism degree of φ with respect to ℬ X and ℬ Y is defined by the following equality:

ISO ( φ ) = BP ( φ ) ∧ BP ( φ − 1 ) .

Theorem 3.19

Let φ : ( X , ℬ X ) → ( Y , ℬ Y ) be a given bijective function, where ℬ X : L X → M and ℬ Y : L Y → M be given mappings. Then we have BP ( φ − 1 ) = BD ( φ ) and so, ISO ( φ ) = BP ( φ ) ∧ BD ( φ ) .

Proof

Since φ : ( X , ℬ X ) → ( Y , ℬ Y ) is a bijective function, it is clear that ( φ − 1 ) ← ( A ) = φ → ( A ) , for each A ∈ L X . This fact implies the following equality:

BP ( φ − 1 ) = ⋀ A ∈ L X ( Born ℬ X ( A ) ↦ Born ℬ Y ( ( φ − 1 ) ← ( A ) ) ) = ⋀ A ∈ L X ( Born ℬ X ( A ) ↦ Born ℬ Y ( φ → ( A ) ) ) = BD ( φ ) .

As a result, it is clear that ISO ( φ ) = BP ( φ ) ∧ BP ( φ − 1 ) = BP ( φ ) ∧ BD ( φ ) . □

Proposition 3.1

Let the mapping ℬ X : L X → M be given, and i d X : X → X be the identity function. Then we have BP ( i d X ) = BD ( i d X ) = ISO ( i d X ) = 1 M .

Proof

The proof is straightforward and therefore omitted.□

Proposition 3.2

Let ℬ X : L X → M , ℬ Y : L Y → M and ℬ Z : L Z → M be given mappings. Let φ : X → Y and ψ : Y → Z be the crisp functions. Then the followings hold.

BP ( φ ) ∧ BP ( ψ ) ≤ BP ( ψ ∘ φ ) .
BD ( φ ) ∧ BD ( ψ ) ≤ BD ( ψ ∘ φ ) .
If φ and ψ are bijective, then ISO ( φ ) ∧ ISO ( ψ ) ≤ ISO ( ψ ∘ φ ) .

Proof

(1) Let φ : X → Y and ψ : Y → Z be given functions. Then the following is satisfied.

BP ( φ ) ∧ BP ( ψ ) = ( ⋀ B ∈ L Y ( Born ℬ Y ( B ) ↦ Born ℬ X ( φ ← ( B ) ) ) ) ∧ ( ⋀ C ∈ L Z ( Born ℬ Z ( C ) ↦ Born ℬ Y ( ψ ← ( C ) ) ) ) ≤ ( ⋀ D ∈ L Z ( Born ℬ Y ( ψ ← ( D ) ) ↦ Born ℬ X ( φ ← ( ψ ← ( D ) ) ) ) ) ∧ ( ⋀ C ∈ L Z ( Born ℬ Z ( C ) ↦ Born ℬ Y ( ψ ← ( C ) ) ) ) = ⋀ D ∈ L Z ( Born ℬ Y ( ψ ← ( D ) ) ↦ Born ℬ X ( ( ψ ∘ φ ) ← ( D ) ) ) ∧ ( ⋀ C ∈ L Z ( Born ℬ Z ( C ) ↦ Born ℬ Y ( ψ ← ( C ) ) ) ) = ⋀ C ∈ L Z ( Born ℬ Y ( ψ ← ( C ) ) ↦ Born ℬ X ( ( ψ ∘ φ ) ← ( C ) ) ) ∧ ( Born ℬ Z ( C ) ↦ Born ℬ Y ( ψ ← ( C ) ) ) = ⋀ C ∈ L Z ( Born ℬ Z ( C ) ↦ Born ℬ X ( ( ψ ∘ φ ) ← ( C ) ) ) = BP ( ψ ∘ φ ) .

(2) and (3) are similarly proved.□

Lemma 3.2

Let ℬ X : L X → M , ℬ Y : L Y → M be two given mappings and φ : X → Y be a bijective function. Then the following characterizations are valid.

BP ( φ ) = ⋀ A ∈ L X ( Born ℬ Y ( φ → ( A ) ) ↦ Born ℬ X ( A ) ) .
BD ( φ ) = ⋀ B ∈ L Y ( Born ℬ X ( φ ← ( B ) ) ↦ Born ℬ Y ( B ) ) .

Proof

(1) Since the function φ : X → Y be a bijective, we have φ ← ( φ → ( A ) ) = A , for each A ∈ L X and φ → ( φ ← ( B ) ) = B , for each B ∈ L Y . Hence, the following implications are obtained.

⋀ A ∈ L X ( Born ℬ Y ( φ → ( A ) ) ↦ Born ℬ X ( A ) ) = ⋀ A ∈ L X ( Born ℬ Y ( φ → ( A ) ) ↦ Born ℬ X ( φ ← ( φ → ( A ) ) ) ) ≥ ⋀ B ∈ L Y ( Born ℬ Y ( B ) ↦ Born ℬ X ( φ ← ( B ) ) ) = BP ( φ ) .

Conversely,

BP ( φ ) = ⋀ B ∈ L Y ( Born ℬ Y ( B ) ↦ Born ℬ X ( φ ← ( B ) ) ) = ⋀ B ∈ L Y ( Born ℬ Y ( φ → ( φ ← ( B ) ) ) ↦ Born ℬ X ( φ ← ( B ) ) ) ≥ ⋀ A ∈ L X ( Born ℬ Y ( φ → ( A ) ) ↦ Born ℬ X ( A ) ) .

These results complete the proof.

(2) It is proved similarly to that of condition (1).□

Theorem 3.20

Let ℬ X : L X → M , ℬ Y : L Y → M be given mappings and φ : X → Y be a bijective function. Then the following characterizations are valid.

ISO ( φ ) = ⋀ A ∈ L X ( Born ℬ X ( A ) ↔ Born ℬ Y ( φ → ( A ) ) ) ISO ( φ ) = ⋀ B ∈ L Y ( Born ℬ Y ( B ) ↔ Born ℬ X ( φ ← ( B ) ) )

Proof

It is straightforward by the previous Lemma.□

Proposition 3.3

Let ℬ X : L X → M , ℬ Y : L Y → M and ℬ Z : L Z → M be given mappings. Let φ : X → Y and ψ : Y → Z be the crisp functions. Then the followings hold.

If φ is surjective, then BP ( φ ) ∧ BD ( ψ ∘ φ ) ≤ BD ( ψ ) .
If ψ is surjective, then BP ( ψ ) ∧ BD ( ψ ∘ φ ) ≤ BD ( φ ) .

Proof

(1) Since the function φ is surjective, it implies that φ → ( φ ← ( B ) ) = B , for each B ∈ L Y . Then it follows that ( ψ ∘ φ ) → ( φ ← ( B ) ) = ψ → ( B ) . Hence, the following inequality holds.

BP ( φ ) ∧ BD ( ψ ∘ φ ) = ⋀ B ∈ L Y ( Born ℬ Y ( B ) ↦ Born ℬ X ( φ ← ( B ) ) ) ∧ ⋀ A ∈ L X ( Born ℬ X ( A ) ↦ Born ℬ Z ( ( ψ ∘ φ ) → ( A ) ) ) ≤ ⋀ B ∈ L Y ( Born ℬ Y ( B ) ↦ Born ℬ X ( φ ← ( B ) ) ) ∧ ⋀ B ∈ L Y ( Born ℬ X ( φ ← ( B ) ) ↦ Born ℬ Z ( ( ψ ∘ φ ) → ( φ ← ( B ) ) ) ) = ⋀ B ∈ L Y ( Born ℬ Y ( B ) ↦ Born ℬ X ( φ ← ( B ) ) ) ∧ ( Born ℬ X ( φ ← ( B ) ) ↦ Born ℬ Z ( ψ → ( B ) ) ) = ⋀ B ∈ L Y ( Born ℬ Y ( B ) ↦ Born ℬ Z ( ψ → ( B ) ) ) = BD ( ψ ) .

(2) It is proved similarly to that of condition (1).□

4 Conclusion

General bornological spaces play a key role in research of convergence in hyperspaces, optimization theory and in study of topologies on function spaces. At present, the theory of bornological spaces is developed in various directions by many authors. To make contribution to the study of fuzzy bornological spaces, we found it reasonable to define the fuzzy bornological degree of a mapping which shows that what extent this mapping is fuzzy bornology. By this way, we obtain the degree of boundedness of a lattice valued fuzzy set in the corresponding bornology. We also described the degree of being bounded and bounded preserving for a mapping defined between fuzzy bornological spaces. We studied their relations and investigated the characterizations of the presented notions.

Acknowledgement

The author would like to express her sincere thanks to the Editor and anonymous reviewers for their most valuable comments and suggestions in improving this article greatly.

Funding information: The author states no funding involved.
Author contributions: The author confirmed the sole responsibility for the conception of the study, presented results, and prepared manuscript.
Conflict of interest: The author states no conflict of interest.

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Received: 2024-05-26

Revised: 2024-11-07

Accepted: 2024-11-29

Published Online: 2024-12-19

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https://doi.org/10.1515/math-2024-0110

Keywords for this article

fuzzy set; bounded L-set; bornological space; bounded mapping

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