Extracting Concepts From Fuzzy Relational Context Families

Fuzzy relational formal concept analysis (FRCA) mines collections of fuzzy concept lattices from fuzzy relational context families, which are special datasets made of fuzzy formal contexts and fuzzy relations between objects of different types. Mainly, FRCA consists of the following procedures: first, an initial fuzzy relational context family is transformed into a collection of fuzzy formal contexts; second, a fuzzy concept lattice is generated from each fuzzy formal context by using one of the techniques existing in the literature. The principal tools to transform a fuzzy context family into a set of fuzzy formal contexts are the so-called fuzzy scaling quantifiers, which are particular fuzzy quantifiers based on the concept of evaluative linguistic expression. FRCA can be applied whenever information needs to be extracted from multirelational datasets including vagueness, and it can be viewed as an extension of both relational concept analysis and fuzzy formal concept analysis. This article contributes to the development of fuzzy relational concept analysis by achieving the following goals. First of all, we present and study a new class of fuzzy quantifiers, called t-scaling quantifiers, to extract fuzzy concepts from fuzzy relational context families. Subsequently, we provide an algorithm to generate, given a t-scaling quantifier, a collection of fuzzy concept lattices from a special fuzzy relational context family, which is composed of a pair of fuzzy formal contexts and a fuzzy relation between their objects. After that, we introduce an ordered relation on the set of all t-scaling quantifiers, which allows us to discover a correspondence among fuzzy concept lattices deriving from different t-scaling quantifiers. Finally, we discuss how the results obtained for t-scaling quantifiers can be extended to the class of fuzzy scaling quantifies. Therefore, this analysis highlights the main differences between t-scaling and fuzzy quantifiers.


I. INTRODUCTION
F ORMAL concept analysis (FCA) is a mathematical theory created to produce a conceptual hierarchy called concept lattice, starting from a formal context, which is a triple composed of a set of objects, a set of attributes, and a relation between objects and attributes [1], [2], [3]. Mathematically, a concept lattice is a particular lattice having formal concepts as elements. B is the set of all attributes of Y being satisfied by all objects of A. According to the philosophical tradition, A and B are, respectively, called extent and intent of the concept. Furthermore, formal concepts are ordered with the subconcept-superconcept relation capturing that a concept can be more specific, or more general, than another (for example, the concept "tiger" is more specific than the concept "feline"). FCA is an appealing research topic from a theoretical perspective [4], [5], [6] and finds applications in different areas of computer science, such as information retrieval, machine learning, and knowledge discovery [7], [8], [9], [10], [11].
A large group of scholars, motivated by the need to solve real-life problems, has extended FCA in several ways (for some examples, see [12], [13], [14], and [15]). In this article, we are interested in fuzzy formal concept analysis (FFCA) and relational concept analysis (RCA). Both are theories proposed to broaden the scope of FCA as follows.
FFCA extends FCA, using fuzzy logic, to also deal with vague information. Shortly speaking, FFCA mines concept hierarchies from datasets called fuzzy formal contexts, where attributes are satisfied by objects with truth degrees belonging to a graded scale, which is usually the real interval [0,1]. Among all the existing FFCA approaches, we focus on the one developed by Bělohlávek [16] and independently by Pollandt [17], where each concept is uniquely determined by a fuzzy set of objects A and a fuzzy set of attributes B connected to each other as follows: given an object x and an attribute y, A(x) is the degree to which x has all attributes of B and B(y) is the degree to which y is shared by all objects of A. Such concepts are constructed by considering complete residuated lattices as algebraic structures of truth degrees [18].
RCA combines FCA with description logic to extract concept hierarchies from multirelational datasets. The RCA input is a relational context family, which is composed of several formal contexts and intercontext relations, namely, relations between objects of different formal contexts. First, the RCA process transforms the initial relational context family in a collection of formal contexts by using the so-called scaling quantifiers. After that, it generates a set of concept lattices (the RCA output) by employing the classical FCA techniques [19], [20], [21].
Scaling quantifiers are binary relations on the power set 2 X of a given universe X and measure how large the intersection of two subsets A and B of X is w.r.t. the size of A. Their definitions carry an existential import, also called presupposition, corresponding to the assumption that the universe of quantification must be nonempty. An example of scaling This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ quantifier is Q 30 : 2 X × 2 X → {0, 1} such that Q 30 (A, B) = 1 if and only if at least 30% of elements in A belong to B, and the intersection between A and B contains at least an element of X (the latter condition represents the existential import). 1 The choice of scaling quantifiers, during the RCA process, is up to one or more users according to the initial dataset and the final classification that they would like to obtain.
Unfortunately, we cannot employ RCA to extract concept hierarchies from vague datasets because RCA only deals with crisp sets and relations. This limit has motivated Boffa et al. [23] to propose a first RCA generalization based on fuzzy logic. Thus, FFCA and RCA have recently been unified to create fuzzy relational concept analysis (FRCA).
FRCA has the purpose of extrapolating information (i.e., collections of fuzzy concept lattices) from multirelational datasets involving vagueness (i.e., fuzzy relational context families). A fuzzy relational context family extends the notion of relational context family by taking into account fuzzy (instead of crisp) relations.
The extraction of fuzzy concept lattices is obtained by employing the so-called fuzzy scaling quantifiers, which are generalizations of RCA scaling quantifiers. Mathematically, fuzzy scaling quantifiers are special fuzzy quantifiers defined on the standard Łukasiewic MV-algebra and based on the concept of evaluative linguistic expressions. These are expressions of natural language having the form hedge big , where an hedge is an adverbial modification like very, extremely, and roughly, and their theory is constructed in a formal system of higher order fuzzy logic (fuzzy type theory) [24], [25], [26]. In addition, fuzzy scaling quantifiers are interpretations in a model of intermediate quantifiers, which are special formulas of the formal theory of intermediate generalized quantifiers presented in [27] and elsewhere.
Let [0, 1] X be the collection of all fuzzy sets on a universe X; an example of fuzzy scaling quantifier is S Very is the truth degree of the sentence "a very big part of A is included in B." 2 In the formula of fuzzy scaling quantifiers, each linguistic expression hedge big is modeled by a function Bi ν : [0, 1] → [0, 1], which is normal and increasing.
In this article, we present and study a new class of FRCA quantifiers called t-scaling quantifiers, which are also extensions of RCA scaling quantifiers. A t-scaling quantifier S t is uniquely determined by a threshold t ∈ [0, 1]. Formally, S t is a function assigning a value of [0,1] to each pair of fuzzy sets of a universe X, where S t (A, B) is the truth degree of the sentence "a part of A being at least as big as t (in the scale [0,1]) is included in B." As for fuzzy scaling quantifiers given in [23], the formula of S t (A, B) includes the subformula x∈X A(x) representing the existential import and capturing that the universe of quantification A must not be empty, i.e., x∈X A(x) is the truth degree of the sentence "there exists at least one element in A." The existential import is a philosophical concept discussed in several publications, especially in those concerning the study of Aristotle square (see [28], [29], [30], [31], and [32] for some examples) Traditionally, it refers to the consideration that the sentence "All As are B" has sense if "As exist." The algebraic structures of truth degrees, chosen to obtain t-scaling quantifiers and the related fuzzy concepts, are complete residuated lattices having [0,1] as support [18]. These are the most used structures in FFCA applications and include the standard Łukasiewicz MV-algebra (already considered in [23]) and the standard Gödel algebra.
In this article, t-scaling quantifiers clearly play a fundamental role. However, generalized quantifiers have recently been introduced in FFCA to extend the definition of concept-forming operators, which are based on the universal quantifier "all" [33], [34], [35], [36].
The main motivations to introduce t-scaling quantifiers in FRCA are explained in what follows. The definition of t-scaling quantifiers is based on a complete residuated lattice having [0,1] as support, which is more general than the standard Łukasiewicz MV-algebra used to define fuzzy scaling quantifiers in [23]. Hence, the concept extraction with t-scaling quantifiers could be realized in future application not necessarily considering the standard Łukasiewicz MV-algebra, but selecting the most appropriate complete residuated lattice [0, 1], ∧, ∨, ⊗, →, 0, 1 , according to the situation to analyze.
Additionally, during the FRCA process, one or more experts in the given domain, who usually do not have mathematical skills, must select the most suitable quantifiers to produce the final concept extraction. Therefore, using t-scaling (instead of fuzzy scaling) quantifiers is convenient for the following reasons.
1) Each t-scaling quantifier is uniquely determined by a threshold belonging to [0,1], while each fuzzy scaling quantifiers by a function from [0,1] to [0,1], which models an evaluative linguistic expression. Therefore, for experts, it is certainly easier to determine thresholds than functions.
2) The meaning of t-scaling quantifiers can be better understood by experts because it can be traced back to the meaning of percentage. Indeed, Theorem 3.4 proves the existence of a one-to-one correspondence between t-scaling quantifiers and scaling quantifiers presented in [22]: for each t ∈ [0, 1], the t-scaling quantifier S t (A, B) is the generalization of the quantifier Q t * 100 (A, B) expressing that at least t * 100% of the elements of A belong to B (the existential import previously described must be included). On the other hand, infinite fuzzy scaling quantifiers forming the classS t * 100 can be viewed as generalizations of Q t * 100 (see Section V-A). Also, S t belongs to the classS t * 100 , when t ≥ 0.5 and we confine to the standard Łukasiewicz MV-algebra (see Remark 3.5). Thus, according to the previous considerations, experts could use S t instead of any quantifier inS t . 3) Experts select the most suitable quantifiers also evaluating how their choice affects the final concept classification. Theorem 4.5 provides a way to compare concept lattices deriving from each pair of different t-scaling quantifiers. Such result helps experts to make the selection according to the final classification that they would like to obtain. Unfortunately, as explained by Remark 5.3, the same is not always possible when we consider a pair of fuzzy scaling quantifiers, and therefore, this makes it more difficult for experts to understand what the best quantifiers to employ are. Although in this article we introduce t-scaling quantifiers, their main results that consist in proposing FRCA algorithms and comparing their corresponding fuzzy concepts are also provided for fuzzy scaling quantifiers.
Essentially, this article extends the study on FRCA started in [23] and aims to provide new tools for data analysis and knowledge discovery in the FCA framework. Furthermore, it responds to the need stated in [22] and other papers to broaden the RCA scope to analyze datasets that involve vagueness. Therefore, the algorithms and results proposed in this article can be applied anytime information needs to be extracted from datasets having the form of fuzzy relational context families. The following is an example. We can consider the fuzzy relations I : where X is a set of individuals; Y is made of personality characteristics like sociable and impulsive; Z is a set of sports like volleyball, yoga, and football; W is made of sport attributes like creative, funny, and aerobic; finally, r expresses how much a given person in X is interested in a given sport in Z. Then, using FRCA and choosing the t-quantifiers with the threshold t = 0.5, we can discover, for instance, that all individuals that are sociable with a degree of at least 0.7 and are interested in at least 50% of sports being both funny and aerobic with a degree of at least 0.8. Furthermore, FRCA can be used to solve all the problems already considered in the RCA applications, but involving datasets characterized by fuzzy relations; for example, the extraction of link key candidate from fuzzy resource description framework (RDF) graphs [37] instead of the classical ones considered in [38] or the construction of fuzzy ontology by extending the results in [39] . The rest of this article is organized as follows. Section II reviews some basic notions and results regarding fuzzy logic and FRCA. Section III defines and studies t-scaling quantifiers, and presents FRCA algorithms. Section IV is devoted to introduce a total order on t-scaling quantifiers and show a correspondence among fuzzy concepts deriving from different t-quantifiers. Then, in Section V, we describe how the results obtained for t-scaling quantifiers in Sections III and IV can be extended to the class of fuzzy scaling quantifies. Therefore, the main differences between t-scaling and fuzzy scaling quantifiers are highlighted. Finally, Section VI concludes this article.

II. PRELIMINARIES
This section focuses on preliminary notions and results we need in this article. Let us underline that all the concepts will be provided by assuming that the initial universe is finite.

A. Mathematical Tools for Fuzzy Logic
is the truth degree of the statement "x belongs to A." In the following, we use the symbol [0, 1] X to denote the collection of all the fuzzy sets of X. Moreover, let A ⊂ ∼ X; we We now review the notion of residuated lattice, which is a general truth structure for fuzzy logic.
The inclusion relation between classical sets is generalized as follows.
Definition II.5: Then, we deal with particular cases of fuzzy measures on fuzzy sets.
Examples of fuzzy measures are defined as follows. Definition II.7 (see [42]): Let AX. Then, the cardinality |A| of A is given by (1) Definition II.8 (see [25]): Let A ⊂ ∼ X. Then, the measure μ A : We require a special operation to form a new fuzzy set from a given one by extracting several elements together with their membership degrees and putting the other membership degrees equal to 0.
Definition II.9 (see [27]): Example II. 10: Why do we need the notion of fuzzy set cuts? In order to provide the formula of the t-scaling quantifier S t (A, B) (see Definition 3.1), we should have considered universes of quantification smaller than A, which correspond to the fuzzy sets included in A according to Definition 2.5. However, the properties of implication suggest considering only the fuzzy sets with membership degrees significantly smaller than those of A. Therefore, a satisfactory solution was to consider the cuts of A, Moreover, we evaluate the size of A|Z w.r.t. A by using the operator Δ t : [0, 1] → [0, 1] that transforms each element of [0,1] being greater than or equal to t in 1 and the remaining ones in 0. Namely, let x ∈ [0, 1]; The concept of inclusion given in Definition 2.5 is generalized as follows.
Definition II. 11: where S X (A, B) represents the degree of inclusion of A in B.
Observe that if S X (A, B) = 1, then A is included in B according to Definition 2.5.
Fuzzy Galois connections and fuzzy closure operators are fundamental notions in fuzzy logic.

B. Fuzzy Formal Concept Analysis
Let [0, 1], ∧, ∨, ⊗, →, 0, 1 be a complete residuated lattice. 3 Definition II.14: A fuzzy formal context is a triple (X, Y, I), where X is a set of objects, Y is a set of attributes, and I is a fuzzy relation on X × Y , i.e., I : Definition II.15 (see [16] and [17]): for all x ∈ X and y ∈ Y . A ↑ I (y) and B ↓ I (x) are the truth degrees of the statements "y is shared by all objects of A" and "x has all attributes of B," respectively.
The following results regarding the operators of Definition 2.15 have been proved in [43], [45], and [46].
Theorem II. 16: Let (X, Y, I) be a fuzzy formal context. Then, the pair made of Theorem II.17: Let (X, Y, I) and (X, Y, J) be fuzzy formal contexts. Then, As shown below, operators of Definition 2.15 are employed to extract fuzzy concepts from every fuzzy formal context. Algorithms for generating all the fuzzy concepts from a fuzzy formal context are provided in [47] and elsewhere.
We denote the set of all fuzzy concepts of (X, Y, I) with B(X, Y, I).
(B(X, Y, I), R) is a complete fuzzy lattice, 4 called the fuzzy concept lattice of (X, Y, I), where the relation R [48].
Theorem II. 19: Let (X, Y, I) be a fuzzy formal context, let A ⊂ ∼ X, and let B ⊂ ∼ Y . Then, A ↑ I ↓ I and B ↓ I ↑ I are, respectively, the extent and the intent of concepts of B((X, Y, I), R).

C. Fuzzy Relational Concept Analysis
In FRCA, a significant role is played by the so-called fuzzy scaling quantifiers, which are generalizations of standard scaling quantifiers by using fuzzy logic.
Among all the RCA scaling quantifiers considered in [22], we are interested in the following.
Definition II.20: Let X be a universe; we put P(X) 2  Scaling quantifiers given by Definition 2.20 have been extended in the fuzzy logic framework as follows (see [23] for more details).
Definition II.21: Let [0, 1], ∧, ∨, ⊗, →, 0, 1 be the standard Łukasiewic MV-algebra; let Bi ν be a function modeling an evaluative linguistic expression with the form hedge big in the context [0,1], and let X be a universe. Then, the fuzzy ν-universal scaling quantifier on X is a function S ν : Remark II. 22: Mathematically, Bi ν is a function from [0,1] to [0,1], which is normal (i.e., there exists at least an element x of [0,1] such that Bi ν (x) = 1) and increasing (i.e., if x ≤ y, then Bi ν (x) ≤ Bi ν (x), for each x ∈ [0, 1]). Bi ν is obtained by composing two functions: Bi modeling the expression Big and ν modeling an adverbial modification called hedge like Very. The role of Bi ν in the previous definition is to evaluate μ A (A|Z). Then, if ν models Very, Bi ν (μ A (A|Z)) is the degree to which the size of A|Z is Very Big w.r.t. the size of A. More explanations are found in [23] and [49].
FRCA analyzes data organized as a fuzzy relational context family.
Definition II. 23 2) A new fuzzy concept lattice is extracted from each fuzzy formal context of K , by using the existing fuzzy FCA techniques. Mainly, step 1 is realized as follows. a) Let SQ be the collection of all fuzzy scaling quantifiers; we consider the functions s : R → SQ, k dom : R → K, and k cod : R → K such that for each fuzzy relation r : A × B → [0, 1]; and k dom (r) and k cod (r) are two fuzzy formal contexts of K having A and B as sets of objects, respectively. 5 b) For each (X, Y, I) ∈ K, we consider the set of relations and let i ∈ {1, . . . , n}; we denote the fuzzy concept lattice extracted by k cod (r i ) with L i . Moreover, given r i : X × Z → [0, 1] and x ∈ X, we use the symbol r i (x) to indicate a fuzzy set of Z such that (r i (x))(z) = r i (x, z) for each z ∈ Z, and the symbols E i C to indicate the extent of the concept C of L i . Then, we construct a new fuzzy formal context (X, Y * , I * ) such that: Therefore, a new family of fuzzy formal context is given by Of course, (X, Y * , I * ) = (X, Y, I) when the set {r ∈ R | k dom (r) = (X, Y, I)} is empty.
Eventually, observe that (X, Y * , I * ) contains both information of (X, Y, Z) and that of the fuzzy relations of {r ∈ R | k dom (r) = (X, Y, I)}.
Then, step 2 can be realized by employing one of the several algorithms introduced in the literature (for example, see [47] and [50]).

III. FRCA WITH T-SCALING QUANTIFIERS
In this section, we first present a new family of fuzzy scaling quantifiers called t-scaling quantifiers (see Section III-A). Subsequently, we show a procedure to mine a collection of fuzzy concept lattices from a special fuzzy relational context family by using a fixed t-scaling quantifier (see Section III-B).

A. T-Scaling Quantifiers
Definition III.1: Let t ∈ [0, 1]. Then, the fuzzy t-scaling quantifier on X is a function S t : [0, 1] X × [0, 1] X → [0, 1] such that, Moreover, S t (A, B) is the truth degree of the following statement: There exists a cut A|Z of A such that "all elements of A|Z belong to B," "there exists at least one element in A|Z," and "the size of A|Z is at least as large as t (in the scale [0,1]) w.r.t. the size of A".
A fundamental role in the definition of t-scaling quantifiers is played by x∈X (A|Z)(x) interpreting the logical formula (∃x)(A|Z)(x) in a model of fuzzy predicate logic. The latter captures that there exists at least one element of X in A|Z and speaks about the existential import (or presupposition). Let us underline that in fuzzy logic, the existential import is included into the formula of quantifiers by the strong conjunction, in order to guarantee the validity of some syllogisms needing the adjunction property [27]. T-scaling quantifiers satisfy the properties shown in the next proposition.  In the following theorem, we show another way to obtain the t-scaling quantifier corresponding to t = 1.
Theorem III.3: Let A, B ⊂ ∼ X; then Proof: By (4) Moreover, since a ∨ 0 = a is satisfied in every bounded lattice, we can conclude that (9) holds. By Theorem 3.3, (8) can be rewritten as follows: Then, S t (A, B) is constructed by applying S 1 to all pairs as (A|Z, B), where A|Z represents a universe of quantification smaller than A, and by using Δ t to evaluate the size of A|Z w.r.t. the size of A.
We can prove that each t-quantifier equals a special RCA scaling quantifier given by Definition 2.20, when both apply to pairs of classical sets of the initial universe.

B. Algorithms in FRCA
This subsection principally provides two algorithms in FRCA. The first one, given t ∈ [0, 1] and A, B ⊂ ∼ X, computes S t (A, B). The second one generates fuzzy concept lattices from a fuzzy relational context family composed of two fuzzy formal contexts (X, Y, I) and (Z, W, J), and a fuzzy relation between X and Z. These algorithms are based on the results presented below.
In the following theorem, we rewrite the formula of S t (A, B) by considering not all, but only specific cuts of A, namely all those whose size w.r.t. A is at least large t.
Theorem III.6: Let A, B ⊂ ∼ X, and let t ∈ [0, 1]; we put Proof: We can rewrite (10) in the following equivalent form: Therefore, the thesis follows from the properties of complete residuated lattices (see Proposition 2.3).
We can find S t (A, B) also by considering in (11) only some of the fuzzy sets in H t (A). To achieve this goal, we need to define and study a family of cuts of A.
Definition III.7: Let A, B ⊂ ∼ X, and let k ∈ K (A,B) , where Remark III.8: It is easy to verify that A k is a cut of A, for each k ∈ K (A,B) . Moreover, The following proposition states that given k ∈ K (A,B) , A k is the maximum on the set of all cuts A|Z of A satisfying a special condition.
. We intend to prove that there exists Z ∈ H * t (A, B) such that S 1 (A|Z, B) ≤ S 1 (A|Z, B).
Thus, we have shown that S 1 (A|Z, B) ≤ S 1 (A|Z, B), where A|Z belongs to H * t (A, B). Hence, using Proposition 2.3(b) again,  (A|Z, B). (A, B), by using Theorem 3.6 again. Now, employing the previous results, we propose the procedure P 1 , which takes as input a pair of fuzzy sets A and B of a universe X, and a threshold t ∈ [0, 1], and finds the value S t (A, B).
In detail, P 1 is based on Theorem 3.10: it computes the supremum of the values corresponding to S 1 (A k , B), where A k is a cut of A given by Definition 3.7 such that μ A (A k ) ≥ t.  procedure We currently have enough tools to present the procedure P 2 . Its input consists of a fuzzy relational context family (K, R), where K = {(X, Y, I), (Z, W, J)} and R = {(X, Z, r)}, and a threshold t ∈ [0, 1], and its output is a pair of fuzzy concept lattices {L 1 , L 2 } associated with (K, R) through S t .
Let us point out that P 2 recalls, in addition to P 1 , the procedures P 3 and P 4 . These, given a fuzzy formal context (X, Y, I), respectively, compute the fuzzy concept lattice of (X, Y, I) and the extent of all fuzzy concepts of (X, Y, I) by using one of the existing FFCA techniques (for example, see [47] and [51]). procedure P 2 ((X, Y, I), (Z, W, J), (X, Z, r), t) Y * ← Y for all x ∈ X do for all y ∈ Y do I * (x, y) ← I(x, y) end for end for

t) {As explained in
Section II-C, r(x) is a fuzzy set such that r(x)(z) = r(x, z)}. end for L 2 ← P 4 (X, Y * , I * ) end for return {L 1 , L 2 } end procedure Eventually, the concept lattices related to a general fuzzy context family (K, R) such that |K| ≥ 2 and |R| ≥ 1 can be obtained by applying the procedure P 2 to (K, K, r, t) for each relation r ∈ R, where t, K, and K are selected, as described in Section II-C.

IV. COMPARING CONCEPT LATTICES DERIVING FROM DIFFERENT T-SCALING QUANTIFIERS
In this section, we first introduce a total order on t-scaling quantifiers. Then, we compare fuzzy concept lattices deriving from different t-scaling quantifiers.
An ordered relation on t-scaling quantifiers can be defined as follows.
Definition IV.1: Let S = {S t | t ∈ [0, 1]}, and let S, S ∈ S. Then The next theorem shows that S is a total order on S, i.e., S S S or S S S, for each S, S ∈ S.
we have and we denote with (X, Y * , I t ) the fuzzy formal context obtained from B(Z, W, J) and (X, Z, r), by using the quantifiers S t . 7 For convenience, we can write ↑ t instead of ↑ I t (dually, ↓ t instead of ↓ I t ). Moreover, given C ∈ B(Z, W, J), the symbol y C indicates the relational attribute associated with C. Remark IV.3: By (7), Theorem 4.2 implies that, given s, t ∈ [0, 1] such that s ≤ t, I t ⊆ I s (i.e., I t (x, y) ≤ I s (x, y) for all x ∈ X and y ∈ Y * ).
Therefore, using Theorem 4.2, we can compare particular fuzzy sets deriving from different t-scaling quantifiers. More precisely, the following proposition holds.
Proof: Let x ∈ X. Since s ≤ t, we have I t (x, y C ) ≤ I s (x, y C ) from Remark 4.3. Consequently, by Proposition 2.3(i), k → I t (x, y C ) ≤ k → I s (x, y C ). Therefore, the thesis follows from Definition 2.15. The next theorem exhibits a connection among fuzzy concepts that are generated by different t-scaling quantifiers. In particular, let s and t be thresholds in [0,1] such that s ≤ t; each fuzzy concept corresponding to t is less than or equal to at least one corresponding to s.
To compare concepts of different lattices, we use an ordered relation on the set We now intend to prove that A ⊆ A * , where A * = (B * ) ↓ s and A = B ↓ t . By Remark 4.3, I t ⊆ I s . Then, by Theorem 2.17, Thus, we can conclude that Let us provide an illustrative example, where concepts arising from different quantifiers are compared through .
Example IV.6: Consider a fuzzy relational context family  Table I . 8 We aim: 1) to find fuzzy concepts hidden in (K, R) using S 0.25 and S 0.75 ; 2) to compare, employing , each fuzzy concept deriving from S 0.75 with at least one deriving from S 0.25 . To achieve these goals, we consider B(Z, W, J) = {C 1 , . . . , C 7 }, where Then, we need to find I 0.75 and I 0.25 , which are fuzzy relations on X × Y * , where Y * = Y ∪ {y C 1 , . . . , y C 7 }, determined by S 0.75 and S 0.25 , respectively. I 0.75 and I 0.25 are defined by Table II and are obtained from B(Z, W, J) and r as follows: given t ∈ {0.25, 0.75} and x ∈ X, Therefore, we can compute the fuzzy concepts of B(X, Y * , I 0.25 ) and B(X, Y * , I 0.75 ), which are listed in Tables III and IV

V. COMPARISON OF T-SCALING AND FUZZY SCALING QUANTIFIERS
LetS be the collection of all the fuzzy scaling quantifiers introduced in [23]. We intend to answer the questions: Can the results obtained for S in the previous sections be extended toS? If so, how?
Let us recall that we need to confine to the standard Łukasiewicz MV-algebra, in order to considerS. Moreover, fuzzy scaling and t-scaling quantifiers substantially differ in their formula: A. Extending Results of Section III to Fuzzy Scaling Quantifiers 1) Proposition 3.2 also holds for the quantifiers ofS. The demonstration can be obtained by substituting Δ t with Bi ν into the proof of Proposition 3.2. This is possible because by Remark 2.22, Bi ν (μ A (A|Z)) ≤ Bi ν (μ A (A)) (i.e., Bi ν is increasing) and Bi ν (μ ∅ (∅|Z)) = 1 (i.e., Bi ν is normal). 2) Regarding Theorem 3.3, we can notice that S 1 ∈S.
In [23], S 1 coincides with the quantifier "all," which is based on the evaluative linguistic expression "utmost" (indicated with Δ 1 ), and it is defined by either (8) or (9). 3) Theorem 3.4 leads to a one-to-one correspondence between Boolean scaling quantifiers given by Definition 2.20 and t-scaling quantifies. In particular, we can consider a bijective function such that Q n → S n 100 for each n ∈ [0, 100] or equivalently its inverse such that S t → Q t * 100 for each t ∈ [0, 1], where by Theorem 3. Such correspondence cannot be replied for the quantifiers ofS by considering that, in general, Theorem 3.3 does not hold for fuzzy scaling quantifiers. Namely, there exists S ∈S \ S that applied on classical sets does not equal any Q n with n ∈ [0, 100]. However, we have proved in [23] that given n ∈ [0, 100], we can find a class of quantifiers S n ⊂S, which is connected with Q n by the following relations: let S ∈S n and Q n ≤ S, and if S(A, B) = 1, then Q n (A, B) = 1. Hence, also quantifiers ofS n can be considered generalizations of Q n . Therefore, the previous considerations suggest us to partition the set of all fuzzy scaling quantifiers as follows: S = n∈[0,100]S n . Furthermore, since both S t and quantifiers ofS t * 100 are generalizations of the Boolean scaling quantifier Q t * 100 , we can identify S t withS t * 100 . 4) Theorem 3.6 can be rewritten for fuzzy scaling quantifiers as follows. Theorem V.1: Let S ν ∈S and let A, B ⊂ ∼ X, we put Proof: The proof is analogous to that of Theorem 3.6. Indeed, we can rewrite (12)  Proof: The proof can be obtained from that of Theorem 3.10 by using the properties of complete residuated lattices. First, we need to substitute everywhere H t (A) and H * t (A, B) with H ν (A) and H * ν (A, B), respectively. Second, in order to prove the inequality Z∈H ν (A) S 1 (A|Z, B) ⊗ Bi ν (μ A (A|Z)) ≤ Z∈H * ν (A,B) S 1 (A|Z, B) ⊗ Bi ν (μ A (A|Z)), the following further sentences must be added after (15). Since (A|Z) (x) ≤ (A|Z)(x) for each x ∈ X and μ A is an increasing function, we get μ A (A|Z) ≤ μ A (A|Z). Moreover, it is true that Bi ν (μ A (A|Z)) ≤ Bi ν (μ A (A|Z)) because Bi ν is an increasing function too. Thus, the inequalities Bi ν (μ A (A|Z)) ≤ Bi ν (μ A (A|Z)) and S 1 (A|Z, B) ≤ S 1 (A|Z, B) imply that S 1 (A|Z, B) ⊗ Bi ν (μ A (A|Z)) ≤ S 1 (A|Z, B) ⊗ Bi ν (μ A (A|Z)). Finally, considering that A|Z ∈ H * ν (A, B), we can conclude that Z∈H ν (A) S 1 (A|Z, B) ⊗ Bi ν (μ A (A|Z)) ≤ Z∈H * ν (A,B) S 1 (A|Z, B) ⊗ Bi ν (μ A (A|Z)). 6) Algorithm 1 can be modified to work with fuzzy scaling quantifiers. Indeed, the procedure P 1 must have the function Bi ν instead of the threshold t as input. Moreover, concerning the if/then statement, we need to substitute the condition μ A (A k ) ≥ t with Bi ν (μ A (A k )) = 0, add l → Bi ν (μ A (A k )) as statement to execute, and write S → S ∪ {(n × m) ⊗ l} instead of S → S ∪ {n × m}. 7) Algorithm 2 can be used for fuzzy scaling quantifiers only by changing the input t of P 2 with Bi ν and the procedure P 1 as explained in the previous point.

B. Extending Results of Section IV to Fuzzy Scaling Quantifiers
The relation given by Definition 4.1 can be extended to the class of fuzzy scaling quantifiers: let S, S ∈S: The results proved in Section IV can be extended for fuzzy scaling quantifiers by take into account S and a specific pair of evaluative linguistic expressions: let S ν 1 , S ν 2 ∈S such that Bi ν 1 ⊆ Bi ν 2 ; then, we have the following.
2) I ν 1 ⊆ I ν 2 , where I ν 1 and I ν 2 are, respectively, related to S ν 1 and S ν 2 by means of (7) (see . Remark V.3: By Theorem 4.2, we can easily consider a total order S on S, namely, (S, S ) is a chain. Then, we can compare the concepts deriving from any pairs of t-scaling quantifiers by using Theorem 4.5. Unfortunately, the same is not possible for the class fuzzy scaling quantifiers by considering that S is not a total order onS. Indeed, let S ν 1 , S ν 2 ∈S; it can happen that S ν 1 S S ν 2 and S ν 2 S S ν 1 . Consequently, we cannot always compare concepts deriving from two different fuzzy scaling quantifiers.

VI. CONCLUSION
In this article, we focused on deriving information (i.e., collections of fuzzy concept lattices) from particular datasets (i.e., fuzzy relational context families) by employing t-scaling and fuzzy scaling quantifiers. As a future project, we intend to introduce and study new quantifiers in fuzzy relation concept analysis. For example, quantifiers extracting negative information from data, i.e., information based on the absence of a certain amount of properties in objects. We would also like to consider and study t-scaling quantifiers as generalized fuzzy subsethood measures by extending the definitions given in [52].
In addition, we will organize special FRCA quantifiers in structures of opposition, similarly to those constructed in [33], [34], and [36]. Moreover, by understanding relationships between FRCA quantifiers of different types, we could discover connections between their derived fuzzy concept lattices.
Finally, we plan to implement the algorithms presented in this article using real datasets and apply our theoretical results to solve concrete problems in other research domains. After that, it would be very interesting to compare, given t ∈ [0, 1], the concept lattices obtained by using the quantifiers ofS t * 100 and the t-scaling quantifiers S t .