Graded concept lattices in fuzzy rough set theory

Abstract

Noting certain limitations of fuzzy concept lattices in rough set theory in terms of their possible applications, we present here a more flexible notion of a graded fuzzy concept lattice in rough set theory. We establish initial facts about the object-oriented version of graded fuzzy concept lattice and illustrate them with some examples, of both theoretical and practical nature. Fuzzy relations, fuzzy rough sets, measure of inclusion, object-oriented fuzzy concepts, graded fuzzy concept lattices in rough set theory Published in Pablo Cordero, Ondrej Kridlo (Eds.): The 16ℎ International Conference on Concept Lattices and Their ∗Corresponding author.

1. Introduction

Formal concept analysis was initiated by R. Wille and B. Ganter [28], [8] in 80-ties of the previous century. The principal subject of the study and at the same time the main tool of research in formal concept analysis are formal concepts and concept lattices. At present concept analysis and corresponding theory of concept lattices is a well developed field of mathematics with many practical applications. The basics of the fuzzy concept analysis and the corresponding theory of fuzzy concept lattices were mainly developed by R. Bĕlohlávek [1], see also [2], etc.

I. Düntch and G. Gediga [7] who worked in the field of model logic introduced approximationtype operators by means of a binary relation. In addition, they used these operators for an alternative notion of a concept lattice, now known as a property-oriented concept lattice, see, e.g. [29], [16] et. al. Afterwards, Y.Y. Yao [ 29] defined an object-oriented concept lattice, in a certain sense dual to the property-oriented concept lattice. Besides, Y.Y. Yao has highlighted the important feature of these lattices, namely, that they can be realized as the analogue of “classic” Wille-Ganter-Bĕlohlávek concept lattices in rough set theory. The fuzzy version of object and property-oriented concept lattices was considered in [6]. In addition, this paper contains deep analysis of relations between the three kinds of (fuzzy) concept lattices, i.e. between WilleGanter-Bĕlohlávek (or WGB for short) (fuzzy) concept lattices, and two (fuzzy) concept lattices in rough set theory, namely the property-oriented and the object-oriented (fuzzy) concept lattices.

Concept lattices, both “classic” and concept lattices in rough set theory, have found many important applications outside pure mathematics. Specifically, the technique based on concept lattices is used in medicine, biology, healthcare, sociology, etc. On the other hand, we know examples of only fragmentary applications of fuzzy concept lattices (of both types) in practice. The reason for this is that the precise matching between extents and intents in fuzzy environment is nearly impossible, see some comments in this concern in Subsection 4.3. To overcome this problem we suggest to replace the notion of a fuzzy concept by a more general notion of a graded fuzzy concept and to develop a more flexible theory of graded fuzzy concept lattices. To realize this approach we start with the notion of a preconcept, viewing it as a certain “potential concept”, and then estimate “conceptuality” of a fuzzy preconcept by means of fuzzy logic tools. In the result we obtain a graded fuzzy concept lattice. We study basic properties of graded fuzzy concept lattices and illustrate them by some specific examples.

In the paper [26] we used similar ideas to develop a graded approach to WGB-fuzzy concept lattices. However, as diferent from graded WGB-fuzzy concept lattices, here we base on forward and backward powerset operators induced by fuzzy relations. As the result the structure of graded fuzzy concept lattices in rough set theory relays on isotone Galois connection (or adjunction) as diferent from WGB-fuzzy concept lattices constructed in accordance with antitone Galois connection.

The paper is structured as follows. In the second, preliminary, section, we recall and refine definitions used in the paper and clarify the framework in which our research is carried out. In the third section forward and backward operators induced by fuzzy relations are defined and used for studying derivation operators, which in turn form the basis of fuzzy concept lattices in fuzzy rough set theory. In section 4, which is central to the article, we develop a gradation approach to concept lattices in fuzzy rough set theory. The fith section is devoted to examples illustrating our approach in particular cases determined by specific choice of a -norm involved in the definition of a fuzzy relation and the choice of fuzzy sets of potential objects and potential properties. In the sixth section we consider some examples of practical nature where graded fuzzy concepts could be useful. In the last, conclusion section some prospects are sketched for the further studies on the basis of this paper.

2. Preliminaries

Lattices, quantales and residuated lattices. We recall here some well known concepts from the theory of lattices see, e.g. [3], [13], [17], in order to clarify the terms used in the paper.

In our paper = (, ≤, ∧, ∨) denotes a complete lattice, that is a lattice in which joins ⋁ and meets ⋀ of all subsets ⊆ exist. In particular 0 ∈ and 1 ∈ are the bottom and the top elements of respectively. A complete lattice is called join-distributive if ∧ (⋁∈ ) = ⋁∈ ( ∧ ) for every ∈ and every { ∣ ∈ } ⊆ . Dually, a complete lattice is called meet-distributive if ∨ (⋀∈ ) = ⋀∈ ( ∨ ). A complete lattice is called bi-distributive if it is join- and meet-distributive.

Let be a complete lattice and ∗ ∶ × → be a binary associative monotone operation. The tuple (, ≤, ∧, ∨, ∗) is called a quantale [21] if ∗ distributes over arbitrary joins: ∗ (⋁∈ ) = ⋁∈ ( ∗ ),(⋁∈ ) ∗ = ⋁∈ ( ∗ ) ∀ ∈ , { | ∈ } ⊆ .

A quantale is integral if the top element acts as the unit, i.e. 1 ∗ = . A quantale is commutative, if ∗ = ∗ for all , ∈ . In what follows by a quantale we mean a commutative integral quantale. A typical example of a quantale is the unit interval endowed with a lower semicontinuous -norm, see, e.g. [15].

In a quantale a further binary operation ↦∶ × → , the residuum, can be introduced as associated with operation ∗ of the quantale (, ≤, ∧, ∨, ∗) via the Galois connection, that is ∗ ≤ ⟺ ≤ ↦ for all , , ∈ .

A quantale (, ≤, ∧, ∨, ∗) provided with the derived operation ↦, that is the tuple (, ≤, ∧, ∨, ∗, ↦) , is known also as a (complete) residuated lattice [17].

In the next proposition we list basic properties of the residuum that can be found in the works of diferent authors, see, e.g. [ 12].

Proposition 1. Basic properties of the residium: ( 1 ) (⋁ ) ↦ = ⋀ ( ↦ ) for all { ∣ ∈ } ⊆ , for all ∈ ; ( 2 ) ↦ ( ⋀ ) = ⋀ ( ↦ ) for all ∈ , for all { ∣ ∈ } ⊆ ; ( 3 ) 1 ↦ = for all ∈ ; ( 4 ) ↦ = 1 whenever ≤ ; ( 5 ) ∗ ( ↦ ) ≤ for all , ∈ ; ( 6 ) ( ↦ ) ∗ ( ↦ ) ≤ ↦ for all , , ∈ ; ( 7 ) ↦ ≤ ( ∗ ↦ ∗ ) for all , , ∈ ; ( 8 ) ∗ ≤ ∧ for any , ∈ ; ( 9 ) ( ∗ ) ↦ = ↦ ( ↦ ) for any , , ∈ .

Fuzzy sets and fuzzy relations. The concept of a fuzzy set was introduced by L.A. Zadeh [31] and then extended to a more general concept of an -fuzzy set by J.A. Goguen [11] where is a complete lattice, in particular a quantale. Given a set its -fuzzy subset is a mapping ∶ → . The lattice and the quantale structure of is extended point-wise to the -exponent of , that is to the set of all -fuzzy subsets of . An -fuzzy relation between two sets and is an -fuzzy subset of the product × , that is a mapping ∶ × → , see, e.g. [27], [32]. An -fuzzy relation is called left connected if ⋀∈ ⋁∈ (, ) = 1 . If for every ∈ there exists ∈ such that (, ) = 1 , then is called strongly left connected . An -fuzzy relation is called right connected if ⋀∈ ⋁∈ (, ) = 1 . If for every ∈ there exists ∈ such that (, ) = 1 , then is called strongly right connected. An -fuzzy relation ∶ × → is called connected (strongly connected) if it is both left and right connected (respectively strongly left and strongly right connected). Since in the paper is an arbitrary but a fixed lattice, we shall omit the prefix and speak just of fuzzy sets and fuzzy relations.

Measure of inclusion of -fuzzy sets. The gradation of a preconcept lattice presented below is based on the fuzzy inclusion between fuzzy sets. We give here a brief introduction into this field.

In order to fuzzify the inclusion relation ⊆ “a fuzzy set is a subset of a fuzzy set ”, we have to interpret it as a certain fuzzy inclusion ↪ based on ”if - then” rule, that is on some implication ⇒ on the lattice . In the result we come to the formula ↪ = inf∈ (() ⇒ ()). As far as we know, this approach was applied for the first time in [ 23], see also [24], where it was based on the Kleene-Dienes implication ⇒. Later the fuzzified relation of inclusion between fuzzy sets was studied and used by many authors, see, e.g. [14], [22], [4], [5], et al. In most of the papers the implication ⇒ was defined by means of residuum ↦ of the underlying quantale (, ≤, ∧, ∨, ∗) .

Definition 1. By setting ↪ = ⋀∈ (() ↦ ()) for all , ∈ , we obtain a mapping ↪∶ × → . We call ↪ by the measure of inclusion of a fuzzy set into the fuzzy set . Let ↩ = ↪ . We denote ≅ = ( ↪ ) ∧ ( ↪ ) and view it as the degree of equality of fuzzy sets and .

Proposition 2. [10]. Properties of the mapping ↪∶ × → : ( 1 ) (⋁ ) ↪ = ⋀ ( ↪ ) for all { ∣ ∈ } ⊆ and for all ∈ ; ( 2 ) ↪ ( ⋀ ) = ⋀ ( ↪ ) for all ∈ , and for all { ∣ ∈ } ⊆ ; ( 3 ) ↪ = 1 whenever ≤ ; ( 4 ) 1 ↪ = ⋀ () for all ∈ where 1 ∶ → is a constant function with the value 1 ∈ ; ( 5 ) ( ↪ ) ≤ ( ∗ ↪ ∗ ) for all , , ∈ ; ( 6 ) ( ↪ ) ∗ ( ↪ ) ≤ ( ↪ ) for all , , ∈ ; ( 7 ) (⋀ ) ↪ (⋀ ) ≥ ⋀ ( ↪ ) for all { ∶ ∈ } , { ∶ ∈ } ⊆ ; ( 8 ) (⋁ ) ↪ (⋁ ) ≥ ⋀ ( ↪ ) for all { ∶ ∈ } , { ∶ ∈ } ⊆ . 3. Forward and backward operators induced by fuzzy relations Let ∶ × → be a fuzzy relation. We refer to the set as the domain and to the set as the codomain of the fuzzy relation . The forward and backward lower and upper operators induced by fuzzy relations can be found in the works of diferent authors and under various names. Since our goal is to apply them in the framework of concept analysis we interpret them here also as derivation operators in the definition of a fuzzy concept.

Definition 2. (□ ) - □ denotes the lower forward operator ⇒ ∶ → . That is, given ∈ and ∈ , let □ ()( ) = □ ( ) = ⋀∈ ((, ) ↦ ()). (♦ ) - ♦ denotes the upper forward operator → ∶ → . That is, given ∈ and ∈ , let ♦ ()( ) = ♦ ( ) = ⋁∈ ((, ) ∗ ()). (■ ) - ■ denotes the lower backward operator ⇐ ∶ → . That is, given ∈ and ∈ , let ■ ()() = ■ () = ⋀∈ ((, ) ↦ ( )). (♦ ) - ♦ denotes the upper backward operator ← ∶ → . That is, given ∈ and ∈ , let ♦ ()() = ♦ () = ⋁∈ ((, ) ∗ ( )).

The proof of the following three propositions is easy and can be found in [25]: Proposition 3. Let , be sets and ∶ × → ( 1 ) 1, 2 ∈ , 1 ≤ 2 ⟹ □1 ≤ □2 ; ♦1 ≤ ♦2 ; ( 2 ) 1, 2 ∈ , 1 ≤ 2 ⟹ 1■ ≤ 2■ ; 1♦ ≤ 2♦ . a fuzzy relation. Then Proposition 4. If a fuzzy relation is strongly left connected, then □ ≤ ♦ . If ∶ × → strongly right connected, then ■ ≤ ♦ . is

Proposition 5.

( 1 ) □ (1 ) = 1 and ♦ (0 ) = 0 . If is left connected, then □ ( ) = ♦ ( ) = for every ∈ . ( 2 ) ■ (1 ) = 1 and ♦ (0 ) = 0 . If is right connected, then ■ ( ) = ♦ ( ) = for every ∈ .

Proposition 6. Let { ∣ ∈ } ⊆ and { ∣ ∈ } ⊆ . Then: ( 1 ) (⋀∈ )□ = ⋀∈ □ ; ( 2 ) (⋁∈ )♦ = ⋁∈ ♦ ; ( 3 ) (⋀∈ )■ = ⋀∈ ■ ; ( 4 ) (⋁∈ )♦ = ⋁∈ ♦ .

Proof. Let { ∣ ∈ } ⊆ and ∈ . Then (⋀∈ □ ) () = ⋀∈ (⋀∈ ((, ) ↦ ()) ) = ⋀∈ (⋀∈ ((, ) ↦ ()) ) = ⋀∈ ((, ) ↦ ⋀∈ () ) = (⋀∈ )□ (). (⋁∈ ♦ ) () = ⋁∈ ♦ () = ⋁∈ (⋁∈ ((, ) ∗ ⋁∈ (⋁∈ ((, ) ∗ ()) ) = ⋁∈ (((, ) ∗

()) ) = ⋁∈ () ) = (⋁∈ )♦ ().

Our next goal is to consider the successive composition of these operators. We denote compositions ♦ ∘ □ = □♦ ; ♦ ∘ ■ = ■♦ , □ ∘ ♦ = ♦□ ■ ∘ ♦ = ♦■ . Respectively, given fuzzy sets ∈ and ∈ , denote □♦ () = □♦ , ♦■ () = ♦■ ; ■♦ () = ■♦ and ♦□ () = ♦□ . ⋀∈ □ () = Theorem 1. For every ∈ and every ∈ it holds: ( 1 ) □♦ ≤ ≤ ■♦ ; ( 2 ) ■♦ ≤ ≤ ♦□ .

Proof. Let ∈ . Then □♦ () = ⋁∈ ((, ) ∗ ⋁∈ ((, ) ∗ ( ⋀ ′∈ (( ′, ) ↦ ( ′))) ≤ ⋁∈ ((, ) ∗ ((, ) ↦ ()) ) ≤ (); on the other hand ♦■ () = ⋀∈ ((, ) ↦ ⋀∈ ((, ) ↦ (⋁ ′∈ ( ′, ) ∗ ( ′))) ≥ ⋀∈ ((, ) ↦ ((, ) ∗ ()) ) ≥ ().

In a similar way the second inequality can be proved. ♦ () ) = □ () ) = Theorem 2. For every ∈ and every ∈ ( 1 ) ♦■ ( 2 ) □♦ = ♦ ; = □ ; ( 3 ) ♦□ ( 4 ) ■♦ = ♦ ; = ■ .

Proof. Applying Theorem 1 we have ♦■ = ( ♦■ )♦ ≥ ♦ ; ♦■ = ( ♦ )■♦ ≤ ♦ and hence ♦■ = ♦ . On the other hand, again by Theorem 1, □♦ = ( □♦ )□ ≤ □ ; □♦ = ( □ )♦□ ≥ □ and hence □♦ = □ . The last two equalities can be proved in a similar way.

From Theorem 2 we get Corollary 1. For every ∈ and every ∈ ( 1 ) ♦■ ( 2 ) □♦ = ♦■ ; = □♦ ; ( 3 ) ♦□ ( 4 ) ■♦ = ♦□ ; = ■♦ , and hence the operators □♦ , ♦■ , ■♦ , ♦□ are idempotent.

Theorem 3. Operators ♦ ∶ → and ■ ∶ → form an adjoint pair ( ♦ , ■ ), that is ♦ ≤ ⟺ ≤ ■ for any ∈ , ∈

.

Operators ♦ ∶ → and □ ∶

→ form an adjoint pair ( ♦ , □ ), that is ♦ ≤ ⟺ □ for any ∈ , ∈

■ assume first that ♦ ≤ . Then by Proposition 3( 2 ) ≤ ■ and by Theorem 1( 1 ) ≤ ■♦

≤ ■ . Conversely, assume that ≤ ■ . Then ♦ ≤ ■♦ by Proposition 3( 1 ) and ♦ ≤ ■♦

≤ by Theorem 1( 2 ).

To show that ♦ ≤ ⟺ ≤ □ assume first that ♦ ≤ . Then by Proposition 3( 2 ) ≤ □ and by Theorem 1( 1 ) ≤ ♦□

≤ □ . Conversely, assume that ≤ □ . Then ♦ ≤ □♦ by Proposition 3( 1 ) and ♦ ≤ □♦

≤ by Theorem 1( 2 ). 4. Degree of conceptuality of fuzzy preconcepts in fuzzy rough sets theory

Definition 3.

called a fuzzy preconcept.

Following the main lines of research in formal concept analysis, we start with a pair of sets and and a fuzzy relation ∶ × →

. Now the set is interpreted as the set of some objects and the set as the set of some properties, the value (, ) ≥ ∈ object has the property at least to the degree . The quadruple ( , , , ) means that the is called a formal fuzzy context.

4.1. Preconcepts and preconcept lattices

Viewing ∈ as a fuzzy set of potential objects and ∈ as a fuzzy set of properties we interpret the pair (, )

as a potential fuzzy concept. These leads to the following definition: [26]1 Given a formal fuzzy context ( , , , ) , a pair = (, ) ∈ × is

To view the set × of all fuzzy preconcepts as a lattice we must first introduce an order on it. To do this reasonably, we must take into account that an increase in the set of objects naturally leads to a decrease in the set of properties that these objects satisfy. Therefore we introduce a partial order ⪯ as follows. Given 1 = ( 1, 1) and 2 = ( 2, 2), we set 1 ⪯ 2 if and only if 1 ≤ 2 and 1 ≥ 2. Let (ℙ, ⪯)be the set Further, given a family of fuzzy preconcepts { = ( , ) ∶ ∈ } ⊆ × , we define its join (supremum) by ⊻∈ = (⋁∈ , ⋀∈ ) and its meet (infimum) as ⊼∈ = (⋀∈ , ⋁∈ ). × endowed with this partial order. 1This notion of a fuzzy preconcept is not related to the notion of a preconcept as it is defined in [ 8] Theorem 4. [26]. (ℙ, ⪯,⊻, ⊼) is a complete lattice. Besides, if is an infinitely bi-distributive lattice, then (ℙ, ⪯,⊻, ⊼) is also an infinitely bi-distributive lattice.

4.2. Concept lattices in rough set theory.

set of properties ∈ In turn, ♦ = { ∈ ∣ Now (, ) ∈ ( , ) Patterned after [ 29], and using our notations, we define an object-oriented fuzzy concept as the pair (, ) ∈ ℙ

such that = defined as a pair (, ) ∈ ♦ and □

= . Similarly, a property-oriented fuzzy concept is × such that = ■ and ♦ = . In what follows, we restrict to considering object-oriented fuzzy concepts; the case of property-oriented fuzzy concepts can be easily obtained just by replacing fuzzy relation ∶ × → Let ( , ) ⊆ × be the family of object-oriented fuzzy concepts. by its inverse −1 ∶ × →

In order to explain the meaning of an object-oriented fuzzy concept, we consider the simplest non-trivial case. Let be a lattice whose all elements are isolated from below, let ∶ × → {0, 1} ⊆ and let ∈ , ∈ . We denote = { ∈ ∣ () ≥ }; = { ∈ ∣ ( ) ≥ } Thus is the set of objects ∈

belonging to at least to degree , respectively is the belonging to at least to degree . Now □ = { ∣ □ ( ) ≥ } can be described as the set of properties ∈ such that if is related to some object , then ∈ = and = ♦ for all ∈ .

which are related to some ∈ . (⋁∈ , (⋁∈ )

■♦ ∈ .

Following the principles of concept analysis, the set (( , ), ⪯) should be realized as a complete lattice, where ⪯ is the relation induced from the partially ordered set (ℙ, ⪯,⊼, ⊻), that is (, ) ⪯ ( ′, ′) if ≤ ′ and ≥

′. Since the coordinate-wise infimum and supremum of (even two) fuzzy concepts need not be a fuzzy concept, we must define infimum and supremum operations on (( , ), ≤)

in a diferent way. We do it patterned after definitions of such operations in case of “classical” (that is Wille-Ganter-Bĕlohlávek) concepts and concepts in rough set theory, see [29]. Namely, given a family = {( , ) ∣ ∈ } ⊆ ( , ) , we define the infimum and the supremum of this family respectively by ) .

Theorem 5. (( , ), ≤,

, ) is a complete lattice.

Proof. We have to show that if = {( , ) ∣ ∈ } ⊆ ( , ) ♦□ , then ((⋀∈ ) , ⋀∈ ) and ) are object-oriented fuzzy concepts. However, we get it immediately by applying statements ( 2 ) and ( 4 ) of Theorem 2 and recalling that = ♦ and □ = for every

4.3. Problems of practical applications of concept lattices.

Unfortunately the scope of practical applications of object-oriented (as well as property-oriented) fuzzy concepts is rather restricted. For example, if ⊆ , ⊆ are crisp sets and ∶ × → then (, ) can be an object-oriented fuzzy concept only in case ♦ = and □ = . This actually means that also must be a crisp relation and hence there is no even any ”flavour” of fuzziness in this case. As a reasonable improvement of this situation we suggest to replace the notion of an (object-oriented, property-oriented) fuzzy concept by a graded (object-oriented, property oriented) fuzzy preconcept. We do it in the next subsection.

4.4. Graded fuzzy concept lattices in rough set theory

As above, let and be sets of objects and properties respectively and ∶ × → be a fuzzy relation. We aim to determine for every pair (, ) ∈ × the extent, or the degree to which this pair (, )

is a fuzzy object-oriented concept (resp. a fuzzy property-oriented concept). This can be done along the lines of our article in [26], which proposes a graded approach in “classical” fuzzy concept analysis.

Definition 4. The degree of (object-oriented) conceptuality of a pair (, ) ∈ by Dob(, ) = ( □ ≅ ) ∧ ( ≅

♦ ).

The degree of (property-oriented) conceptuality of a pair (, ) ∈ ( ♦ ≅ ) ∧ ( ≅ ■ ).

× is defined × is defined by Dpr(, ) = By changing pairs (, )

over × we obtain operators Dob ∶ × → and Dpr ∶ × → . ) ∧ ( ≅ ) ∧ ( ≅ ∈ .

Definition 5.

A graded fuzzy object-oriented concept lattice is the lattice ℙ endowed with operator Dob ∶ × → , that is the tuple (ℙ, ⪯,⊻, ⊼, Dob).

A graded fuzzy property-oriented concept lattice is the lattice ℙ endowed with operator Dpr ∶ × → , that is the tuple (ℙ, ⪯,⊻, ⊼, Dpr).

Going on with the study of graded fuzzy concept lattices in rough set theory, we shall restrict to the case of an object-oriented lattice (ℙ, ⪯,⊻, ⊼, Dob), that is when Dob(, ) = ( ♦ ). The corresponding results for the property-oriented lattice Dpr(, ) = ( ♦ ) can be obtained in a similar way just by replacing fuzzy relation ∶ × → by the fuzzy relation −1 ∶ × → defined by −1(, ) ∶= ( , ) for all ∈ , for all □

≅ □ ≅

The computational complexity of the operator Dob ∶ × → laid in the definition of the graded fuzzy object-oriented concept lattice (ℙ, ⪯,⊻, ⊼, Dob) and multidirectional components involved in this definition are the reason for further study of these operators by dividing them into several components. 4.5. Operators D1ob, D2ob, D3ob, D4ob ∶ × → Given a pair (, ) ∈ ℙ D4ob(, ) = ↩ ♦ . Obviously, let D1ob(, ) = □ ↪ , D2ob(, ) = □ ↩ , D3ob(, ) = ↪ ♦ , Dob(, ) =

D1ob(, ) ∧

D2ob(, ) ∧

D3ob(, ) ∧

D4ob(, ).

Next, we study the properties of the operators D1ob, D2ob, D3ob, D4ob separately. Theorem 6. Properties of operator D1ob ∶ ℙ( , , , ) →

Proof. Referring to Proposition 6 and Proposition 2 we have D1ob (⋀∈ ( , )) = D1ob (⋀∈ , ⋁∈ )) = (⋀∈ )□ ⋀∈ □ ↪ ⋁∈ ≥ ⋀∈ ( □ ↪ ) ≥ ⋀∈ D1ob( , ).

From the definition it is clear that D1ob(, 1 ) = □ applying Proposition 5 we have D1ob(0 , ) = 0 □

3 Theorem 7. Properties of operator Dob ∶ ℙ( , , , ) → 1. Operator D1ob ∶ ℙ( , , , ) → is upper semicontinuous, that is

1 1

Dob (⋀∈ ( , )) ≥ ⋀∈ Dob( , ) ∀{( , ) ∶ ∈ } ⊆ ℙ( , , , ). 2. D1ob(, 1 ) = 1 for every ∈ . In particular D1ob(0 , 1 ) = 1. If is left connected, then 1

Dob(0 , ) = 1 for every ∈ . 1. Operator D3ob ∶ ℙ( , , , ) → is upper semicontinuous, that is

3 3

Dob (⋀∈ ( , )) ≥ ⋀∈ Dob( , ) ∀{( , ) ∶ ∈ } ⊆ ℙ( , , , ). 2. D3ob(0 , ) = 1 for every ∈ in particular, D3ob(0 , 1 ) = 1. If is right connected, then 3

Dob(, 1 ) = 1 for every ∈ .

Proof. Referring to Proposition 6, Proposition 2 we have ⋀∈ D3ob (⋀∈ ( , )) = D3ob (⋀∈ , ⋁∈ )) = ⋀∈ ↪ (⋁∈ )♦ = ⋀∈ ↪ ⋁∈ ♦ ≥ ⋀∈ ( ↪ ♦ ) ≥ ⋀∈ Dob( , ).

3 Just from definitions we have D3ob(0 , ) = 0 ↪ ♦ = ⋀∈ (1 ↦ ♦ ( )) ≥ 1 ↦ 1 = 1. In case is right connected we apply Proposition 5 and have Dob(, 1 ) = ↪ 1 ♦ = ↪ 1 = 3 (( ) → 1) = 1 Theorem 8. Properties of operator D2ob ∶ ℙ( , , , ) → 1. Operator D2ob ∶ ℙ( , , , ) → is lower semicontinuous, that is

2 2 Dob (⋁∈ ( , )) ≥ ⋀∈ Dob( , ) ∀{( , ) ∶ ∈ } ⊆ ℙ( , , , ).

2 2 2. Dob(, 0 ) = 1 for every ∈ and Dob(1 , ) = 1 for every ∈ , in particular 2 Dob(1 , 0 ) = 1.

↪ ⋁∈ = ↪ 1 ≥ 1 ↦ 1 = 1. If is leftconnected, then ↪ = 0 ↪ = ⋀∈ (0 ↦ ( )) = 1.

.

Theorem 9. Properties of operator D4ob ∶ ℙ( , , , ) → 1. Operator D4ob ∶ ℙ( , , , ) → is lower semicontinuous, that is

4 4 Dob (⋁∈ ( , )) ≥ ⋀∈ Dob( , ) ∀{( , ) ∶ ∈ } ⊆ ℙ( , , , ).

4 4 2. Dob(, 0 ) = 1 for every ∈ and Dob(1 , ) = 1 for every ∈ . In particular 4 Dob(1 , 0 ) = 1; ≥ ⋁∈ ↩ ⋀∈ ♦

≥ ⋀∈ ( ↩ ♦ ) = ⋀∈ (0 ↦ ()) = 1 .

Proof. Referring to Proposition 6, Proposition 2 we have

♦ D4ob (⋁∈ , ⋀∈ ) = ⋁∈ ↩ (⋀∈ )

4 ⋀∈ Dob( , ).

Applying Proposition 5 D4ob(, 0 ) = ↩ 0 ♦ = 0♦ ↪ = 0 ↪ = In turn D4ob(1 , ) = 1 ↩ ♦ = ♦ ↪ 1 = ⋀∈ ( ♦ () ↦ 1) = 1 .

Corollary 2. If fuzzy relation is connected, then: • Operators D1ob, D3ob ∶ × → are upper semicontinuous and D1ob(0 , ) = D3ob(0 , ) = 1 for every ∈ and D1ob(, 1 ) = D3ob(, 1 ) = 1 for every ∈ . • Operators D2ob, D4ob ∶ × → are lower semicontinuous and D2ob(, 0 ) = D4ob(, 0 ) = 1 for any ∈ and D2ob(1 , ) = D4ob(1 , ) = 1 for any ∈

5. Examples of graded fuzzy concept lattices

Example 1. Let and be sets partitioned into three disjoint subsets = Further, let relation ∶ × → {0, 1} be defined by In this section, we present two theoretical examples of graded fuzzy concept lattices. The first of these examples as well as its generalizations (obtained by employing fuzzy sets and with diferent number of values) of this example will be used for illustration of possible applications outside “pure” mathematics. One of such applications is presented in the next subsection. The second is a quite “artificial”, example. It is added in order to illustrate the fundamental role of the fuzzy relation involved in the definition of graded fuzzy concepts and to stress the importance of the reasonable choice of fuzzy relation when considering special situations. 2 2 ⋃=0 and = ⋃=0 . (, ) = - for the Łukasiewicz -norm Dob(, ) = 1− ∣ − ∣

Example 2. Let and be sets partitioned into three disjoint subsets = and let ∶ → [0, 1] relation ∶ × → {0, 1} be defined as in the previous example. Further, let and hence Dob(, ) = 0 .

country risks 6. Application of graded fuzzy concept lattices in assessment of We consider the imposition of international sanctions (embargoes) to diferent countries for illustration of possible application of graded fuzzy concept lattices. These sanctions are introduced either by international organizations (like the United Nations) or diferent countries (including the European Union) to restrict economic activities of particular countries posing diferent global risks via prohibiting the trading of certain goods or provision of particular services, limiting the access to financial resources, etc. Sanctions are also applied to diferent private individuals and legal entities. Consequently, several countries and their residents are subject to restrictive measures of lower or higher degree which can be mathematically illustrated using fuzzy sets. Meanwhile only North Korea is considered as fully sanctioned country, but the recent emergence of the war in Ukraine has induced a global wave of new sanctions against Russia making it as almost fully sanctioned country with only few leftovers for very limited non-restricted trades with commodities, e.g. natural gas.

In order to set up a conceptual lattice reflecting this matter we consider the countries as objects and the sanctions as properties. The contents of this lattice can be visualized, e.g. using data from the site www.sanctionsmap.eu comprising information on sanctions imposed by the European Union and the United Nations, and these data can be further enriched by diferent other sanctions imposed by countries with major global influence, like the United States of America. Naturally, based on our Example 1 provided in the previous section all non-sanctioned countries are included in the subset 0, all fully sanctioned countries are included in the subset 2, and all partly sanctioned countries are included in the subset 1 thus creating the set of objects. For the sake of clarity we should admit that terms ”partly” and ”fully” are conditional and depend on the total number of particular sanctions applied to corresponding countries. Furthermore we introduce the set of properties by applying diferent importance to particular sanctions. In such case the subset 0 is introduced as subset not containing any sanctions, subset 2 contains the sanctions imposing the strongest possible restrictions, e.g. prohibition of any financial transactions, and subset 1 contains all other types of sanctions targeting particular sectors of economic activities. For the purposes of our example we have chosen the value of equal to 0.4 and the value of equal to 0.6. We obtain the following fuzzy sets and and corresponding values for □ and ♦ :

The meaning of □ is explained as relation of particular subsets of sanctions to corresponding countries while the meaning of ♦ is explained as relation of particular subsets of countries to corresponding sanctions. Furthermore we obtain the following values of Dob(, ) : 0.8 in case of Łukasiewicz -norm, 0.67 in case of product -norm and 0.4 in case of minimum -norm. These results basically demonstrate that the level of country sanctioning, i.e. ”partial” or ”full” cannot exceed the importance of sanctions applied to respective countries. This is also the mathematical reasoning for, e.g. dificulties in taking decision on disconnection of all Russian ifnancial institutions from the global network of the Society for Worldwide Interbank Financial Telecommunication (SWIFT) while still enabling of at least limited trades with commodities (e.g. natural gas). This particular example was used to demonstrate the applicability of concept lattice for analysis of international sanction in a very simplified way. It is obvious that in a more realistic scenario we must refine both the partitioning of the set of objects (countries) and the set of properties (sanctions). As a result we would be dealing with more granulated setup, disjoint subsets =

⋃=0 and corresponding values of 1, ..., and 1, ..., . These values of and

are still finite and in practice do not exceed the number of oficially recognized countries (currently there are 197 countries) and the number of meaningful sanctions. The analysis of such more granulated landscape is subject to our further research.

7. Conclusion

Operators D1ob, D2ob, D3ob and D4ob ∶ × → were introduced in order to evaluate various aspects of the correspondence between a fuzzy set of (potential) objects and a fuzzy set of (potential) properties. Based on these operators, we defined graded fuzzy concept lattices and studied their main properties. Our next research will focus on using these operators for the construction of so called, conceptional hulls of fuzzy preconcepts. Given a fuzzy preconcept (, ) ∈ × and ∈ , the triple (, ↑, ↓), where ↓ ≤ ≤ ↑, is called the concept -hull of a fuzzy preconcept (, ) , if

1 (, ↑) ∧ 3 (, ↑) ≥ and 2 (, ↓) ∧ 4 (, ↓) ≥ and in addition ↑, ↓ are the “optimal” fuzzy sets satisfying these conditions. We shall carry out the study of the construction × ⇒ × × assigning to each (, ) ∈ × its conceptional -hull (, ↑, ↓). Specifically we plan to investigate algebraic (in particular categorical) and topological properties of this construction. In addition, we intend to apply this construction for practical purposes, in particular, for the choice of strategy developed in response to the risks related to global political and economic developments.

The second interesting direction, kindly indicated us by a referee, is the study of the relations between graded fuzzy concept lattices and pattern structures described by certain semi-lattices, see, e.g. [9], [18]. In such structures patterns are used instead of attributes (properties), allowing thus a certain gradation of concepts.

Acknowledgments

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