Constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges

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Constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges Lubomir Antoni ∗ , Stanislav Krajˇci, Ondrej Krídlo Institute of Computer Science, Faculty of Science, Pavol Jozef Šafárik University in Košice, Jesenná 5, 040 01, Košice, Slovakia Received 20 January 2015; received in revised form 7 December 2015; accepted 13 December 2015

Abstract The paper deals with the isomorphism between the constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges. The essential point of the former approach encompasses the full diversification of data structures within a formal context. In particular, we use a different complete lattice for diverse objects, a different complete lattice for diverse attributes and a different poset for diverse matrix fields. The latter framework with heterogeneous hedges results in a reduction in the size of the corresponding concept lattice. We present the properties of constraint heterogeneous approach that is associated with the fixpoints of hedges and we add remarks to the related studies. © 2015 Elsevier B.V. All rights reserved. Keywords: Formal concept analysis; Algebra; Heterogeneous concepts; Hedges

1. Introduction Formal concept analysis [23] scrutinizes an object-attribute block of the binary relational data under the notion of a formal concept. The classical approach mines the formal concepts that are generated from the binary relation and several fuzzifications have been proposed [6,13,41,42]. A fuzzy setting considers a degree to which an object has a particular attribute and in particular, the extensions of L-fuzzy concept lattices encompass generalized concept lattices [28], multi-adjoint concept lattices [32,35,21], Galois connectional concept lattices [39,40], interval-valued L-fuzzy contexts [1]. The extensions taking into account the structures of idempotent semifields and their completions [43] or multiply diversification of structures [3] are another answer to generalizations in formal concept analysis. Concerning a categorical approach, [30,31] present the intercontextual relationships of formal contexts. Formal concept analysis, in general, is an interesting research area that provides theoretical foundations, fruitful methods, algorithms and underlying applications in many areas and has been investigated in relation to various disciplines [18]. Krajˇci [29] proved that generalized concept lattices cover concept lattices with hedges [9,10]. Generalized concept lattices are based on three sets of truth degrees as a kind of a three-sorted residuated structure with a specific isotone * Corresponding author. Tel.: +421 55 234 2541.

E-mail addresses: [email protected] (L. Antoni), [email protected] (S. Krajˇci), [email protected] (O. Krídlo). http://dx.doi.org/10.1016/j.fss.2015.12.007 0165-0114/© 2015 Elsevier B.V. All rights reserved.

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aggregation function [7]. Heterogeneous hedges within a context are introduced by Bˇelohlávek and Vychodil [11] as the extension for L-fuzzy formal context with the same hedge for the set of objects and the same hedge for the set of attributes [9,10]. The reduction in size of the concept lattice is demonstrated and the theorem which shows that stronger hedges lead to a smaller concept lattice is proved. On the other hand, isotone fuzzy Galois connections related to L-fuzzy formal context with hedges are studied by Koneˇcný [25]. Non-commutative conjunctors from fuzzy logic programming [36–38] are the cornerstone for the framework of multi-adjoint concept lattice [35]. In [33], Medina and Ojeda-Aciego introduce the notion of L-connected lattices. L-connection of two complete lattices generalizes the notion of hedges in formal concept analysis. In [26], Koneˇcný et al. describe that the selection of complete sup-semilattice preserves extents of the original multi-adjoint L-connected concept lattice and it is shown how their results generalizes the propositions about subcontexts from [23]. The construction of adjoint triples using hedges and the conditions for their generation are formulated in [26]. In our paper, we would like to clarify the relationship between the heterogeneous concept lattices [3] and the novel extension of concept lattices with heterogeneous hedges given by Bˇelohlávek and Vychodil [11]. Our aim is to continue our research on heterogeneous data in formal concept analysis and to fill in the gaps of our previous studies. First, we investigate the properties of the constraint heterogeneous concept lattices with respect to hedges. A translation process between fuzzy setting and ordinary crisp setting provides a way of stating relationship between heterogeneous concept lattices and concept lattices with heterogeneous hedges. We state that Butka et al. in [15] describe a representation of fuzzy concept lattices (represented as crisp complete lattices) in the framework of classical concept lattices, as well. They involve the principal ideal as the structure for a translation process from fuzzy formal context into a binary formal context. On the contrary, we work with an ordinary subset of the Cartesian product defined directly from the particular fuzzy membership function. Moreover, we create the heterogeneous concept lattice first and then find a representation of this heterogeneous concept lattice as a classical concept lattice, therefore the principal ideal can be omitted in our paper. In particular, Section 2 provides preliminaries on concept lattices with heterogeneous hedges. Section 3 describes the extended observations about the interpretation of hedges in formal concept analysis and investigates the properties of fuzzy membership functions with respect to translation process. Section 4 recalls the basic notions of our heterogeneous approach; Section 5 explains how to construct the isomorphism on constraint heterogeneous concept lattices. Remarks on related studies are included in Section 6. We argue about the university admission tests represented by the constraint heterogeneous concept lattice in a working example in Section 7. Finally, conclusions are summarized. 2. Concept lattices with heterogeneous hedges Formal concept analysis with heterogeneous hedges is intensively studied in [11]. This new approach has been introduced by Bˇelohlávek and Vychodil. A new environment extends the one from [9,10]. We remind the basic notions. Let L be a set of truth degrees, then L, ∨, ∧, ⊗, →, 0, 1 forms a complete residuated lattice in which L, ∨, ∧, 0, 1 is a complete lattice with 0 and 1 being the least and greatest element of L, L, ⊗, 1 is a commutative monoid, ⊗ and → satisfy adjointness. Consider a set of objects X, a set of attributes Y and a binary fuzzy relation R between X and Y such that R(x, y) ∈ L. We introduce a definition of a hedge from [10], which is close to a logical connective from [24]. Definition 1. (See [10].) A hedge is a unary mapping ∗ on L such that for each a, b ∈ L: 1∗ = 1, a ∗ ≤ a, (a → b)∗ ≤ a ∗ → b∗ , a ∗∗ = a ∗ . Denote the set of all fixpoints of ∗ in L by fix(∗) = {a ∈ L : a ∗ = a}.

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The fuzzy concept-forming operators with the hedges in a basic form are studied in [9,10]. For L-fuzzy membership functions (or L-fuzzy sets) f ∈ LX and g ∈ LY put  f ↑ (y) = (f (x)∗X → R(x, y)), x∈X

↓

g (x) =



(g(y)∗Y → R(x, y)),

y∈Y

where ∗X , ∗Y are hedges on L as defined above. An extended environment [11] considers two systems of hedges. We will denote systems of hedges by X = (∗x : L → L, x ∈ X) and Y = (∗y : L → L, y ∈ Y ) in accordance with indexed collection of hedges in [11]. An extended approach [11] utilizes the modification of concept-forming operators for heterogeneous type of hedges given by  f ↑ (y) = (f (x)∗x → R(x, y)), x∈X

↓

g (x) =



(g(y)∗y → R(x, y)).

y∈Y

The previous modification of the mappings ↑ and ↓ by heterogeneous hedges requires an introduction of the following operators IX : LX → LX , IY : LY → LY defined: (IX (f ))(x) = (f (x))∗x , (IY (g))(y) = (g(y))∗y and it is worth to investigate their properties. In order to express that, we will use the notation fxa ∈ LX for a singleton function f ∈ LX such that:  a if x = x, a fx (x ) = 0 elsewhere for x, x ∈ X, a ∈ L. For a set of attributes, the definition of singleton function is analogous. Theorem 1. (See [11].) There is a one-to-one correspondence between a system X = (∗x , x ∈ X) and an operator IX satisfying: IX (fx1 ) = fx1 , IX (f ) ⊆ f, IX (f → g) ⊆ IX (f ) → IX (g), IX (IX (f )) = IX (f ),    fxax = IX (fxax ), IX x∈X

x∈X

for each a, b ∈ L and f, g ∈ LX . The correspondence is defined by: (IX (f ))(x) = (f (x))∗x

and

a ∗x = (IX (fxa ))(x).

Analogous correspondence is given by Y = (∗y , y ∈ Y ) and IY . This correspondence gives an answer how to apply two systems of different hedges X , Y to f ∈ LX and g ∈ LY , respectively. From [11] we have that for {ai ∈ L : i ∈ I } and for {bi ∈ L : i ∈ I } it holds  ∗x    ∗y   ∗ ∗ ai∗x = ai∗x bi y = bi y . and i∈I

i∈I

i∈I

i∈I

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  In a special case, if i∈I ai = aj and i∈I bi = bj for some j ∈ I , then from [26] one can write  ∗x    ∗y   ∗ ai = ai∗x bi = bi y . and i∈I

i∈I

i∈I

i∈I

The set HHCL(X, X , Y, Y , R, L) = {f, g : f ↑ = g, g ↓ = f } of all fixpoints (↑, ↓) is called a fuzzy concept lattice with heterogeneous hedges. In [11], a relationship between fuzzy concept lattices with heterogeneous hedges and ordinary concept lattices is proved, whereby Cartesian representation of L-fuzzy membership functions is utilized. Such representation was introduced independently by [4,41], however these mappings play an important role in [5,27], as well. To remind this notation, L-fuzzy membership functions f ∈ LX and g ∈ LY one can represent as a subset of Cartesian product by f L = {x, a ∈ X × L : a ≤ f (x)}

and

gL = {y, b ∈ Y × L : b ≤ g(y)}

and conversely, the subsets of Cartesian product C ⊆ X × L and D ⊆ Y × L determine L-fuzzy membership functions given by:   CL (x) = {a ∈ L : x, a ∈ C} and DL (y) = {b ∈ L : y, b ∈ D}. In order to describe the relationship between fuzzy concept lattices with heterogeneous hedges and ordinary concept lattices, it is needed to formally describe Cartesian representation with respect to heterogeneous hedges. More formally, for X = (∗x , x ∈ X) and Y = (∗y , y ∈ Y ), one can take the fixpoints of hedges and to construct the Cartesian representation of these hedges directly by: X × = {x, a ∈ X × L : a ∈ fix(∗x )}

and

Y × = {y, b ∈ Y × L : b ∈ fix(∗y )}.

In other words, X × is a subset of X × L which contains pairs whereby the second component is a fixpoint of ∗x for all x ∈ X. Analogously for Y × (see [11]). Finally, take the arbitrary subsets C ⊆ X × L and D ⊆ Y × L. The second components of these subsets can be modified with respect to the hedges. For this reason, the operators X and Y are introduced in [11]: C X = {x, a ∗x  : x, a ∈ C}

and

D Y = {y, b∗y  : y, b ∈ D}.

It is worth to see that C X ⊆ X × and D Y ⊆ Y × . In this way, Theorem 2 serves the important result for our paper. Theorem 2. (See [11].) The concept lattices HHCL(X, X , Y, Y , R, L) with heterogeneous hedges are isomorphic to the ordinary concept lattices B(X × , Y × , R × ), where R × is a ordinary relation between X × and Y × corresponding to a Galois connection of the mappings  : X × → Y × and  : Y × → X × defined by ↑

C  = CL LY

↓

D  = DL LX .

and

The ordinary relation R × is given by: x, a, y, b ∈ R ×

iff

a ⊗ b ≤ R(x, y).

As a consequence, the corresponding ordinary concept lattice obeys the main theorem of concept lattices as it is investigated in [23]. For more details we refer to [9–11]. Hájek in [24] and Esteva et al. in [22] deal with the axiomatization of hedges as the unary logical connectives vt, called very true, in a logical approach. 3. Extended observations In general, the natural interpretation of a hedge is based on the truth-stressing of the linguistic expressions or formulas. For instance, the truth degree of formula “a student has a good performance” is higher than or equal to a truth degree of formula “a student has a very good performance”. In what follows, we give an interpretation of hedges applied on truth degrees of fuzzy membership functions in formal concept analysis. A percentile (or a centile) is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. For example, if a score is in the 76th percentile, it is higher than 76%

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Table 1 The hedges in basic and heterogeneous form applied on students. Notation

Interpretation

x f (x) f (x)∗X f (x)∗x

a student a percentile rank of the test score of a student a general classification level (A!,A–E,FX) a national classification level

Fig. 1. A system of heterogeneous hedges applied on a subset of Cartesian product in a graphical representation.

of the other scores. For instance, a degree f (x) = 0.76 in Table 1 means that a test score is greater than or equal to 76% of the scores of students taking the test. The 76th percentile can imply a degree f (x)∗X = 0.7 expressing the general classification level C for 70th–79th percentile. The special case of the 100th percentile corresponds to the classification level A!. Given two students x1 , x2 , a percentile rank f (x1 ) = 0.76 of Slovakian student corresponds to a degree f (x1 )∗x1 = 0.7 that means classification level C (70th–79th percentile) in Slovakian ranking. However, the same degree f (x2 ) = 0.76 of German student can imply a degree f (x2 )∗x2 = 0.75 that means classification level B (75th–89th percentile) in relation with German indicators. Regarding the athletes, their scores and classification limits, interpretation of hedges is similar. In order to make this paper self-contained and since we will use it frequently in a translation from fuzzy concept lattices to the crisp concept lattices, we include basic facts about the mappings alluded to the previous section. Lemma 1. (See [10,29].) For f ∈ LX and g ∈ LY : f L L = f

gL L = g.

and

Lemma 2. (See [10,29].) For C ∈ L × X and D ∈ L × Y : CL L ⊇ C

DL L ⊇ D.

and

Lemma 3. (See [11].) For f ∈ LX and g ∈ LY : f LX L = IX (f )

and

gLY L = IX (g).

We include Fig. 1 to illustrate the different notions related with the hedges and the mappings L and L applied on a fuzzy membership function and a subset of Cartesian product, respectively. Fig. 2 illustrates the previous lemma in a numerical way regarding a system of hedges X = (∗x1 , ∗x2 ) and fuzzy membership function f , whereby f (x1 ) = 0.75, f (x2 ) = d. Notice that f LX L = f does not hold in general. The definition of commutative Galois connection one can see in [4]. In this paper, we present the extended fact that the pair ,  forms an ordinary Galois connection which is not commutative (with respect to  LL ). In fact, it holds for C ∈ X × that ↑

Y × CL  L = CL L L L ∈ Y ,

but on the other side, in general, holds ↑

C  L L = CL LY L L ∈ Y × L.

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Fig. 2. A system of heterogeneous hedges applied on a subset of Cartesian product in a numerical way.

In the following, we explore the mappings  and  (i.e. they are acting directly on the fixpoints of heterogeneous hedges) and we show that such mappings have some additional properties in comparison with L and L (i.e. the mappings which are acting on a residuated lattice L). Let F be a set of all functions f with a domain X such that f (x) ∈ fix(∗x ) for all x ∈ X. Let G be a set of all functions g with a domain Y such that g(y) ∈ fix(∗y ) for all y ∈ Y . First, notice that one can write:  X × = {x, a ∈ X × L : a ∈ fix(∗x )} = ({x} × fix(∗x )) x∈X

and



Y × = {y, b ∈ Y × L : b ∈ fix(∗y )} =

({y} × fix(∗y )).

y∈Y

Then, for f ∈ F and g ∈ G define an ordinary subset of the Cartesian product by f  = {x, a ∈ X × : a ≤ f (x)}

and

g = {y, b ∈ Y × : b ≤ g(y)}

and conversely, for C ⊆ X × and D ⊆ Y × define:   C(x) = {a ∈ fix(∗x ) : x, a ∈ C} and D(y) = {b ∈ fix(∗y ) : y, b ∈ D}. Note that the properties of L and L are satisfied for  and , as well. On the contrary, there exists some properties of  and  which do not hold in general for L. Some of the properties of modified mappings  and  with respect to the operators IX , IY and X , Y are shown in the following lemma. Lemma 4. Let f ∈ F , g ∈ G, S ∈ X × , T ∈ Y × . Then it is true that a) IX (S) = S and IY (T ) = T , b) f X  = IX (f ) = f and gY  = IY (g) = g. Proof. By sequel we have a) (IX (S))(x) ∗x = (S(x))  = ( {a ∈ fix(∗x ) : x, a ∈ S})∗x

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Fig. 3. Fixpoints of the heterogeneous hedges.

 = {a ∈ fix(∗x ) : x, a ∈ S} = S(x) For IY (T ) = T  analogously. b)  f X (x) = {a ∈ fix(∗x ) : x, a ∈ f X } = {a ∈ fix(∗x ) : x, a ∈ f } = ( {a ∈ fix(∗x ) : x, a ∈ f })∗x = ((f )(x))∗x = IX (f )(x) = IX (f )(x) = f (x) (f ∈ F and IX is idempotent) For gY  = IY (g) = g analogously. 2 Fig. 3 illustrates the second part of previous lemma in a numerical way using two sets of fixpoints fix(∗x1 ), fix(∗x2 ) and fuzzy membership function f ∈ F , whereby f (x1 ) = 1, f (x2 ) = e. The basic properties of modified mappings  and  with respect to operations of supremum, infimum and/or union, intersection are proved in the following lemma. Lemma 5. For i ∈ I let fi ∈ F , gi ∈ G, Si ∈ X × , Ti ∈ Y × . Then it is true that a) b) c) d) e)



i∈I fi  =  i∈I fi  and i∈I gi  =  i∈I gi ,  i∈I fi  = i∈I fi and  i∈I gi  = i∈I gi , i∈I fi  =  i∈I fi  and i∈I gi  =  i∈I gi ,  i∈I fi  =

i∈I fi and  i∈I gi  =

i∈I gi ,  i∈I Si  ≤ i∈I Si  and  i∈I Ti  ≤ i∈I Ti .

Proof. a) x, a ∈



i∈I fi 

iff (∀i ∈ I ) : x, a ∈ fi  iff (∀i ∈ I ) : a ≤ fi (x)

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iff a ≤ ( i∈I fi )(x) iff x, a ∈  i∈I fi 

For i∈I gi  =  i∈I gi  analogously.  b)   i∈I gi (y)  = {b ∈ fix(∗y ) : y, b ∈ i∈I gi } = {b ∈ fix(∗y ) : (∃i ∈ I ) : y, b ∈ gi } = {b ∈ fix(∗y ) : (∃i ∈ I ) : b ≤ gi (y)} = ( i∈I gi )(y)   For  i∈I fi  = i∈I fi analogously.  c) x, a ∈ i∈I fi  iff (∃i ∈ I ) : x, a ∈ fi  iff (∃i ∈ I ) : a ≤ fi (x) iff a ≤ ( i∈Ifi )(x) iff x, a ∈  i∈I fi    For i∈I gi  =  i∈I gi  analogously. d)   i∈I gi (y) = {b ∈ fix(∗y ) : y, b ∈ i∈I gi } = {b ∈ fix(∗y ) : (∀i ∈ I : y, b ∈ gi } = {b ∈ fix(∗y ) : (∀i ∈ I : b ≤ gi (y)} = ( i∈I gi )(y)

For  i∈I fi  = i∈I fi analogously. e)   i∈I Si (x) = {a ∈ fix(∗x ) : x, a ∈ i∈I Si } = {a ∈ fix(∗x ) : (∀i ∈ I )x, a ∈ Si } ≤ {a ∈ fix(∗x ) : (∀i ∈ I )a ≤ {t ∈ fix(∗x ) : x, t ∈ Si }} = {a ∈ fix(∗x ) : (∀i ∈ I )a ≤ Si (x)}} = i∈I Si (x).

For  i∈I Ti  ≤ i∈I Ti  analogously. 2 On the other hand, we state that a representation of fuzzy formal contexts in the classical setting in a different manner is described by Butka et al. in [15]. The construction of a binary formal context is based on the principle ideal, not by an ordinary subset of the Cartesian product given by particular fuzzy membership function as in this paper. 4. Heterogeneous concept lattices We introduce the notions of the heterogeneous formal context and the heterogeneous concept lattices [3] which extend the classical approach in formal concept analysis and its notation [23]. Consider a set of objects X, a set of attributes Y . Let P = ((Px,y , ≤Px,y ) : x ∈ X, y ∈ Y ) be a system of posets and let R be a function from X × Y such that R(x, y) ∈ Px,y for all x ∈ X and y ∈ Y . Let U = ((Ux , ≤Ux ) : x ∈ X) and V = ((Vy , ≤Vy ) : y ∈ Y ) be systems of complete lattices. (For simplicity, we omit the indices for all ≤? , since it is always clear which one is used.) Let  = (•x,y : x ∈ X, y ∈ Y ) be a system of operations such that •x,y is from Ux × Vy to Px,y and is isotone and left-continuous in both arguments, that is: 1a) a1 ≤ a2 implies a1 •x,y b ≤ a2 •x,y b for all a1 , a2 ∈ Ux and b ∈ Vy , 1b) b1 ≤ b2 implies a •x,y b1 ≤ a •x,y b2 for all a ∈ Ux and b1 , b2 ∈ Vy ,

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 2a) If a •x,y b ≤ p for some b ∈ Vy , p ∈ Px,y and all a ∈ W ⊆ Ux , then W • x,y b ≤ p, and 2b) If a •x,y b ≤ p for some a ∈ Ux , p ∈ Px,y and all b ∈ Z ⊆ Vy , then a •x,y Z ≤ p. Then we call the tuple X, Y, P, R, U , V,  a heterogeneous formal context. Note that if Ux = Vy and •x,y is commutative, then these conditions can be reduced to the following two: 1) a1 ≤ a2 implies a1 •x,y b ≤ a2 •x,y b for all a1 , a2 , b ∈ Ux = Vy , and  2) If a •x,y b ≤ p for some b ∈ Ux = Vy , p ∈ Px,y and all a ∈ W ⊆ Ux = Vy , then W •x,y b ≤ p. Let F be a set of all functions f with a domain X such that f (x) ∈ Ux for all x ∈ X (more formally, F = x∈X Ux ). Let G be a set of all functions g with a domain Y such that g(y) ∈ Vy for all y ∈ Y (i.e., G = y∈Y Vy ). We define the mapping  : F → G. If f ∈ F , then (f ) ∈ G is defined by  ((f ))(y) = {b ∈ Vy : (∀x ∈ X)f (x) •x,y b ≤ R(x, y)}. Symmetrically, we define the mapping  : G → F . If g ∈ G, then (g) ∈ F is defined as  ((g))(x) = {a ∈ Ux : (∀y ∈ Y )a •x,y g(y) ≤ R(x, y)}.  (, ) are defined by applying the so-called ( , •x,y )-product. The properties of such a structure are described in the literature [7,19]. A construction of a heterogeneous concept lattice follows. Theorem 3. (See [3].) Let f ∈ F and g ∈ G. Then the following conditions are equivalent: 1. f ≤ (g). 2. g ≤ (f ). 3. f (x) •x,y g(y) ≤ R(x, y) for all x ∈ X and y ∈ Y . Corollary 1. Mappings  and  form a Galois connection. We call a pair f, g from F × G such that (f ) = g and (g) = f a heterogeneous concept. Lemma 6. (See [3].) If f1 , g1  and f2 , g2  are heterogeneous concepts, then f1 ≤ f2 iff g1 ≥ g2 . The ordering of heterogeneous concepts is defined by f1 , g1  ≤ f2 , g2  iff f1 ≤ f2 (or equivalently g1 ≥ g2 ). Note that Lemma 6 assures the equivalence of ordering of heterogeneous concepts via ordering of extents and ordering of intents. Finally, we call the poset of all concepts ordered by ≤ a heterogeneous concept lattice, denoted by HCL(X, Y, P, R, U , V, ). The following theorem shows that a heterogeneous concept lattice is in fact a complete lattice with the meet and join operators given as follows. For a direct version of the self-contained proof and for the corresponding isomorphisms in the basic theorem we refer to [3]. Theorem 4 (Basic theorem on heterogeneous concept lattices). (See [3].) 1. A heterogeneous concept lattice HCL(X, Y, P, R, U , V, ) is a complete lattice in which

    fi , gi  = fi ,   gi i∈I

and

i∈I

i∈I

     fi , gi  =   fi gi . , i∈I

i∈I

i∈I

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Table 2 The structures of the constraint heterogeneous concept lattice. Heterogeneous formal context Px,y X Y Ux Vy •x,y Dom(R) = X × Y R(x, y) ∈ Px,y

Constraint heterogeneous formal context x ∈ X, y ∈ Y

L X Y fix(∗x ) fix(∗y ) ⊗ Dom(R) = X × Y R(x, y) ∈ L

x ∈X y ∈Y x ∈ X, y ∈ Y x ∈ X, y ∈ Y

x∈X y∈Y x ∈ X, y ∈ Y

2. For each x ∈ X and y ∈ Y , let Px,y have the least element 0Px,y such that 0Ux •x,y v = u •x,y 0Vy = 0Px,y for all u ∈ Ux , v ∈ Vy . Then a complete lattice Lis isomorphic to HCL(X, Y, P, R, U, V, ) if and only if there are mappings α : x∈X ({x} × Ux ) → L and β : y∈Y ({y} × Vy ) → L such that: 1a) α does not increase in the second argument (for a fixed first argument); 1b) β does not

decrease in the second argument (for a fixed first argument); 2a) Rng(α) is -dense in L1 ; 2b) Rng(β) is -dense in L; and 3) For every x ∈ X, y ∈ Y , a ∈ Ux and b ∈ Vy , α(x, a) ≥ β(y, b)

if and only if

a •x,y b ≤ R(x, y).

Relationships between the related studies are referred in [3,2]. We remind that Medina and Ojeda-Aciego present an approach based on heterogeneous conjunctors [33,34]. A generalization that works with the one-sided heterogeneous structures is introduced in [16,14]. 5. Constraint heterogeneous concept lattices In this section, we consider a special case of heterogeneous concept lattice. We remind that the hedges from [10] are defined on complete residuated lattices (the third condition in Definition 1 requires residuum). In the following lemmas, we assume a more general version of hedges defined on the complete lattice L, where the third condition is loosened to a ≤ b implies a ∗ ≤ b∗ for a, b ∈ L, so the hedge is a kernel operator on L which preserves unit. Lemma 7. (See [29,11].) If a ≤ b then a ∗ ≤ b∗ for a, b ∈ L. Moreover, it is proved in [29] that if W ⊆ fix(∗), then



W ∈ fix(∗). Then, one can obtain that

Lemma 8. (See [29].) fix(∗) is a complete lattice. In general, a subset of a lattice L can be a lattice without being a sublattice. One can see from [26] that fix(∗) is a complete ∨-subsemilattice of L. The previous remarks allow us to replace two systems of complete lattices U , V by (fix(∗x ), x ∈ X), (fix(∗y ), y ∈ Y ) in a definition of a heterogeneous formal context. The translation process from the heterogeneous formal context to the constraint heterogeneous formal context is described in what follows. Left part of Table 2 corresponds to the underlying structures of heterogeneous approach, the right part indicates the constraints of these structures with respect to the concept lattices with heterogeneous hedges described in Section 2. Dom(R) denotes the mapping domain of R. 1 Rng(α) denotes the mapping range of α.

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The special case of heterogeneous formal context allows us to constrain the corresponding concept-forming operators  and  in a following way. Let F be a set of all functions f with a domain X such that f (x) ∈ fix(∗x ) for all x ∈ X. Let G be a set of all functions g with a domain Y such that g(y) ∈ fix(∗y ) for all y ∈ Y . The mapping  : F → G is defined by  ((f ))(y) = {b ∈ fix(∗y ) : (∀x ∈ X)f (x) ⊗ b ≤ R(x, y)}. Symmetrically, the mapping  : G → F is defined by  ((g))(x) = {a ∈ fix(∗x ) : (∀y ∈ Y )a ⊗ g(y) ≤ R(x, y)}. Now, we describe the isomorphism between the corresponding constraint heterogeneous concept lattice (denoted shortly as CHCL) and an ordinary concept lattice (shortly as B) introduced in the previous section. First, we formulate isomorphism Φ and show that Φ is defined correctly. Then we define its inverse Ψ and prove that Ψ is also defined correctly. Consequently, in the second part we investigate composition of the mappings Φ and Ψ . Finally, we prove that Φ and Ψ are order-preserving. I. Part: definition and correctness of Φ and its inverse Ψ Let f, g ∈ CHCL such that f (x) ∈ fix(∗x ) and g(y) ∈ fix(∗y ) for all x ∈ X and y ∈ Y . Then, it is necessary to hold that Φ(f, g) ∈ B. Consider Φ(f, g) = f , g. Hence, we will show that f , g ∈ B, it means that f  = g and f  = g for f  ∈ X × and g ∈ Y × . Observe that f  = f ↑ LY = f ↑ LY (from Lemma 2) = {y, b∗y  ∈ Y × : y, b ∈ f ↑ L } = {y, b ∈ Y × : y, b ∈ f ↑ L } (if b ∈ fix(∗y ), then b∗y = b) ↑ = {y, b ∈ Y × : b ≤ f

(y)} × = {y, b ∈ Y : b ≤ x∈X ((IX (f ))(x) → R(x, y))} = {y, b ∈ Y × : (∀x ∈ X)(IX (f ))(x) ⊗ b ≤ R(x, y)} = {y, b ∈ Y × : (∀x ∈ X)f (x) ⊗ b ≤ R(x, y)} (seeing that f (x) ∈ fix(∗x )) and on the other hand g = {y, b ∈ Y × : b ≤ g(y)} = {y, b ∈ Y × : b ≤ ((f  ))(y)} = {y, b ∈ Y × : b ≤ {t ∈ fix(∗y ) : (∀x ∈ X)f (x) ⊗ t ≤ R(x, y))}}  and since left-continuity {y, b ∈ Y × : (∀x ∈ X)f (x) ⊗ b ≤ R(x, y))} and {y, b ∈ Y × : b ≤ {t ∈ fix(∗y ) : (∀x ∈ X)f (x) ⊗ t ≤ R(x, y))}} equal. Thus, f  = g and the second part of assertion f  = g can be checked dually. Let S, T  ∈ B such that S ⊆ X× and T ⊆ Y × . Then, it is necessary to check that Ψ (S, T ) ∈ CHCL. Consider Ψ (S, T ) = S, T . Hence, we will show that S, T  ∈ CHCL, it means that (S) = T  and S = (T ) for S(x) ∈ fix(∗x ) and T (y) ∈ fix(∗y ) for all x ∈ X and y ∈ Y . Observe that for all y ∈ Y : (S)(y)  = {b ∈ fix(∗y ) : (∀x ∈ X)S(x) ⊗ b ≤ R(x, y)} ∈ X)(IX (S))(x) ⊗ b ≤ R(x, y)} = {b ∈ fix(∗y ) : (∀x

= {b ∈ fix(∗y ) : b ≤ x∈X ((IX (S))(x) → R(x, y))} = {b ∈ fix(∗y ) : b ≤ S↑ (y)} = {b ∈ fix(∗y ) : y, b ∈ S↑ L } = {b ∈ fix(∗y ) : y, b ∈ S↑ LY } (if b ∈ fix(∗y ), then b∗y = b) and on the other side

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T (y) = {b ∈ fix(∗y ) : y, b ∈ T } = {b ∈ fix(∗y ) : y, b ∈ S  } = {b ∈ fix(∗y ) : y, b ∈ S↑ LY }. Thus, (S) = T  and it holds T (y) ∈ fix(∗y ). The second part of assertion given by S = (T ) whereby S(x) ∈ fix(∗x ) can be checked dually. II. Part: composition of Φ and its inverse Ψ Let f, g ∈ CHCL such that f (x) ∈ fix(∗x ) and g(y) ∈ fix(∗y ) for all x ∈ X and y ∈ Y . Then, it is necessary to check that Φ(Ψ (f, g)) = f, g, it means that f , g = f, g. This equality is obtained directly by Lemma 2. Let S, T  ∈ B such that S ⊆ X × and T ⊆ Y × . Then, it is necessary to check that Ψ (Φ(S, T )) = S, T , it means that S, T  = S, T . Observe that y, b ∈ T  iff b ≤ T  (y) iff b ≤ {t ∈ fix(∗y ) : y, t ∈ T } iff b ≤ {t ∈ fix(∗y ) : y, t ∈ S  } iff b ≤ {t ∈ fix(∗y ) : y, t ∈ S↑ LY } iff b ≤ {t ∈ fix(∗y ) : y, t ∈ S↑ L } (if t ∈ fix(∗y ), then t ∗y = t ) iff b ≤ {t ∈ fix(∗y ) : t ≤ S↑ } iff b ≤ S↑ (y) ∧ b ∈ fix(∗y ) iff y, b ∈ S↑ L ∧ b ∈ fix(∗y ) iff y, b ∈ S↑ LY (if b ∈ fix(∗y ), then b∗y = b) iff y, b ∈ S  iff y, b ∈ T and S = S can be proved dually. III. Part: order-preserving of Φ and its inverse Ψ Let f1 , g1  ∈ CHCL and f2 , g2  ∈ CHCL such that f1 (x) ∈ fix(∗x ), f2 (x) ∈ fix(∗x ) and g1 (y) ∈ fix(∗y ), g2 (y) ∈ fix(∗y ) for all x ∈ X and y ∈ Y . Moreover, f1 , g1  ≤ f2 , g2 , that means f1 (x) ≤ f2 (x) and g1 (y) ≥ g2 (y) for all x ∈ X and y ∈ Y . Thus, it is necessary to check that Φ(f1 , g1 ) ≤ Φ(f2 , g2 ), it means that f1 , g1  ≤ f2 , g2 . Observe that f1 , g1  ≤ f2 , g2  iff (f1 ≤ f2 ) ∧ (g1 ≥ g2 ) thus ({x, a ∈ X × : a ≤ f1 (x)} ⊆ {x, a ∈ X × : a ≤ f2 (x)}) ∧ ({y, b ∈ Y × : b ≤ g1 (y)} ⊇ {y, b ∈ Y × : b ≤ g2 (y)}) iff (f1  ⊆ f2 ) ∧ (g1  ⊇ g2 ) iff f1 , g1  ≤ f2 , g2 . Let S1 , T1  ∈ B and S2 , T2  ∈ B such that S1 ⊆ X × , S2 ⊆ X × and T1 ⊆ Y × , T2 ⊆ Y × . Moreover, S1 , T1  ≤ S2 , T2 , that means S1 ⊆ S2 and T1 ⊇ T2 . Thus, we will check that Ψ (S1 , T1 ) ≤ Ψ (S2 , T2 ), it means that S1 , T1  ≤ S2 , T2 . Observe that S1 , T1  ≤ S2 , T2  iff (S1 ⊆ S2 ) ∧ (T  1 ⊇ T2 ) thus (∀x ∈ X : ( {a ∈ fix(∗x ) : x, a ∈ S1 } ≤ {a ∈ fix(∗x ) : x, a ∈ S2 })) ∧ (∀y ∈ Y : ( {b ∈ fix(∗y ) : y, b ∈ T1 } ≥ {b ∈ fix(∗y ) : y, b ∈ T2 })) iff ((∀x ∈ X) : S1 (x) ≤ S2 (x)) ∧ ((∀y ∈ Y ) : T1 (y) ≥ T2 (y)) iff (S1  ≤ S2 ) ∧ (T1  ≥ T2 ) iff S1 , T1  ≤ S2 , T2 .

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Fig. 4. Relationship between heterogeneous concept lattices and lattices with heterogeneous hedges.

Note that every order-isomorphism between complete lattices is automatically a lattice-isomorphism [23,20], i.e. Φ and Ψ preserves all existing joins and meets. The following theorem holds in conclusion: Theorem 5. The constraint heterogeneous concept lattices denoted by CHCL = CHCL(X, Y, F, G, L, R, ⊗) and the ordinary concept lattices B = B(X × , Y × , R × ) are isomorphic and the isomorphisms are: Φ(f, g) = f , g and Ψ (S, T ) = S, T , where f ∈ F , g ∈ G and S ⊆ X × , T ⊆ Y × . In comparison with results presented in [11], here one can omit  and write in the formulation of the isomorphism S instead of S and dually T  instead of T  . As a consequence of the Theorem 2 and Theorem 5 we obtain that constraint heterogeneous concept lattices and concept lattices with heterogeneous hedges are canonically isomorphic. Since constraint heterogeneous concept lattices represent a special case of the heterogeneous concept lattices, we have that an every concept lattice with heterogeneous hedges is canonically isomorphic with some heterogeneous concept lattice (as shown in Fig. 4). 6. Remarks on related studies Our results provide the continuation of the ideas which appear mainly in four related articles [11,33,26,15]. In comparison with these studies, our added value reflects the fact that not only conjunctors can be heterogeneous, but each particular object and each particular attribute disposes of some unique fixpoint of a hedge (the idea for heterogeneous hedges is from Bˇelohlávek and Vychodil [11], described in our Section 2). Moreover, we prove that the extension from [11] is a special case of our heterogeneous approach. The proof of similar transformation is given by Butka et al. [15], but they do not use the fixpoints of hedges in their assumptions. Thus, we collect all the substantial ideas from the mentioned papers and moreover, we work with one system of hedges on the object side and one (another) system of hedges on the attribute side. In particular, Butka et al. [15] describe the representation of fuzzy concept lattices of heterogeneous character [39] in the framework of ordinary concept lattices. They involve a principal ideal in a translation process from fuzzy formal context into a binary formal context. For any complete lattice L, the principal ideal for the element a ∈ L is generated by id(a) = {a ∈ L : a ≤ a}. On the contrary, we work with an ordinary subset of the Cartesian product defined directly from the particular fuzzy membership function. In this direction, the first prospects about the transformation of L-fuzzy concept lattices to ordinary concept lattices using the Cartesian presentation were advocated by Bˇelohlávek in [4], but this transformation does not provide heterogeneous character of objects and attributes. Medina and Ojeda-Aciego [33] provide the construction of L-connected concept lattices (including the heterogeneous conjunctors). Consider complete lattices (L, ), (L1 , 1 ), (L2 , 2 ). Then L1 and L2 are called L-connected if there exist non-decreasing mappings ψ1 : L1 → L, φ1 : L → L1 , ψ2 : L2 → L and φ2 : L → L2 satisfying φ1 (ψ1 (a)) = a and φ2 (ψ2 (b)) = b for all a ∈ L1 , b ∈ L2 . The mappings φ1 and φ2 can be viewed as two hedges for objects and attributes, respectively. Medina and Ojeda-Aciego introduce a multi-adjoint frame which gathers the heterogeneous conjunctors and the sets of truth degrees for objects, attributes and table fields. Moreover, they present a multi-adjoint context (X, Y, R, σ ), whereby σ is a mapping which associates any element from X with some particular

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Table 3 Characterization of related studies. Approach

Carriers for set of objects (or attr.)

Carriers for conjunctors

Type of conjunctors

Type of reduction (or connection)

Bˇelohlávek et al. [11] Medina et al. [33] Koneˇcný et al. [26]

homog. homog. homog.

homog. heterog. heterog.

homog. heterog. heterog.

Butka et al. [15] our approach

heterog. heterog.

heterog. heterog.

heterog. heterog.

heterog. hedges homog. L-connection homog. L-connection and homog. hedges no heterog. hedges

adjoint triple (⊗, , ); for more results on adjoint triples, see [19]. Then the operators ↑c: LX → LY , ↓c : LY → LX are given for all f ∈ LX and g ∈ LY by   R(x, y) σ (x) φ1 (f (x)) f ↑c (y) = ψ2 x∈X

  g (x) = ψ1 R(x, y) σ (x) φ2 (g(y)) . ↓c

y∈Y

Operators ↑c and ↓c have properties which provide the definition of concept, even though they are not closure operators and do not form a Galois connection. This framework generalizes concept lattices with hedges in a basic form [10,29], since L-connection can be viewed as a more abstract notion of the hedges. Medina and Ojeda-Aciego proved that multi-adjoint L-connected concept lattices [33] and multi-adjoint concept lattices [35] are isomorphic. Koneˇcný et al. [26] import the specific properties of hedges from L-fuzzy concept lattices to the framework of multi-adjoint L-connected concept lattices. In particular, they describe a tool for reducing multi-adjoint L-connected concept lattices by selection of complete ∨-subsemilattices. If complete ∨-subsemilattices are replaced by fixpoints of hedges on L1 , L2 and the restrictions of heterogeneous conjunctors to these fixpoints are taken, the resulting multi-adjoint concept lattices are isomorphic to those which are generated by applying the hedges to multi-adjoint L-connected concept-forming operators. Moreover, Koneˇcný et al. [26] show how to generate new adjoint triples by applying the hedges to conjunctors and implications; and prove that by taking new adjoint triples in a multi-adjoint frame, the multi-adjoint concept lattices are isomorphic with two previously described constructions. Our results demonstrate that the transformations remain feasible even if the multiple heterogeneity is considered (Table 3). The last column indicates that to cover all the substantial features from Table 3, we can connect all the input carriers from constraint heterogeneous approach using the so-called heterogeneous L-connection. Consider complete lattices (fix(∗x ), ), (fix(∗y ), ) and (L, ) such that fix(∗x ) and fix(∗y ) are L-connected for every pair x, y ∈ X × Y . Let F be a set of all functions f with a domain X such that f (x) ∈ fix(∗x ) for all x ∈ X. Let G be a set of all functions g with a domain Y such that g(y) ∈ fix(∗y ) for all y ∈ Y . Then we can obtain the concepts in which the sets of attributes and objects are evaluated in the same carrier by the following concept-forming operators  : F → G and  : G → F:   ((f ))(y) = ψy {b ∈ fix(∗y ) : (∀x ∈ X)φx (f (x)) ⊗ b ≤ R(x, y)} ,  ((g))(x) = ψx {a ∈ fix(∗x ) : (∀y ∈ Y )a ⊗ φy (g(y)) ≤ R(x, y)}, whereby ψx : fix(∗x ) → L, ψy : fix(∗y ) → L, φx : L → fix(∗x ) and φy : L → fix(∗y ) are non-decreasing mappings for every pair x, y ∈ X × Y verifying that φx (ψx (a)) = a, and φy (ψy (b)) = b for all a ∈ fix(∗x ) and b ∈ fix(∗y ). 7. Worked example In this paper, we deal with a formal context with heterogeneous hedges, an ordinary formal context and a constraint heterogeneous formal context. Fig. 5 illustrates the examples for these three extensions, whereby concept lattices of these examples are isomorphic. The heterogeneous approach provides a way how to formulate the specific preferences (individual and/or group preferences), for example see [3]. The hedges one can use to reduce the sets of truth degrees. In what follows, we

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Fig. 5. Formal context with heterogeneous hedges, ordinary formal context and constraint heterogeneous formal context.

would like to declare an example of the constraint heterogeneous approach given on a university admission testing. Consider two different universities for objects and two different standardized tests (Math and Computer Science, theoretical and practical, etc.) for attributes. Each university uses different criteria for admission. In our constraint heterogeneous formal context from Fig. 5, the degrees of objects2 stand for: admission to university (1); conditionally admission to university (1/3); rejection to university (0). Degrees of attributes include: a passed admission test (0); a partially passed admission test (2/3); a failed admission test (1). Degrees of table fields express a coefficient (or threshold) determining the possible cases for admission to university. The value 1 indicates that all cases are possible. By concept-forming operators of constraint heterogeneous approach, in which Łucasiewics conjunctors are applied, one can reveal the following intuitive observations from our example: (1, 0) = 0, 2/3 – to be admitted to the first university, I have to pass the first test and partially pass the second test, (0, 1) = 1, 0 – to be admitted to the second university, I can fail the first test, but I have to pass the second test, (1/3, 0) = 0, 1 – to be conditionally admitted to the first university, I have to pass the first test, but I can fail the second test, (1, 1) = 0, 0 – to be admitted to both universities, I have to pass both tests, (1, 1) = 0, 0 – if I failed both tests then I will be rejected by university, (0, 2/3) = 1, 0 – if I passed the first test and partially passed the second test then I will be admitted only to the first university. Notice that L-connection from [33] can be viewed here as an effort to unify the admission degrees of all universities, or classification degrees of all tests. 8. Conclusions and future work This paper focuses on a particular relationship between two general approaches to the theory of formal concept analysis. Specifically, between the so-called heterogeneous concept lattices (HCL) and the concept lattices with het2 To improve readability throughout the section, we write 1 instead of 1 fix(∗x1 ) and (1, 0) instead of (1fix(∗x1 ) , 0fix(∗x2 ) ). The same

holds for the fixpoints of attributes.

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erogeneous hedges (HHCL). We have investigated the special case of heterogeneous concept lattices with respect to the hedges. In this paper, truth-stressing hedges were considered. Another possibility is to apply truth-depressing hedges as in [25]. Moreover, an exploration of the fuzzy subsethood degrees and their properties serve a worthwhile topic for future work with respect to constraint heterogeneous concept lattices or, in general, within the framework of heterogeneous concept lattices. The problem is whether every Galois connection can be formulated by our concept lattice construction. An interesting point for future work is to explore isomorphism of concept lattices associated with the two data tables as it is done in [8] on Gödel fuzzy logic connectives. Multilattices as the underlying sets of truth values are studied in [17]. The framework that enables the user to change the level of granularity of attributes [12] inspires us for our future work. In conclusion, our proposed extension provides an efficient way to represent heterogeneous data and we will explore this uncharted territory in formal concept analysis. Acknowledgements We would like to thank the reviewers for their comments and remarks which were important to improve the presentation and the comprehensibility of the manuscript. This work was supported by the Scientific Grant Agency of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences under contract VEGA 1/0073/15. References [1] C. Alcalde, A. Burusco, R. Fuentes-González, I. Zubia, The use of linguistic variables and fuzzy propositions in the L-fuzzy concept theory, Comput. Math. Appl. 62 (8) (2011) 3111–3122. [2] L. Antoni, S. Krajˇci, O. Krídlo, B. Macek, L. Pisková, Relationship between two FCA approaches on heterogeneous formal contexts, in: L. Szathmary, U. Priss (Eds.), Proceedings of the 9th International Conference on Concept Lattices and Their Applications, 2012, pp. 93–102. [3] L. Antoni, S. Krajˇci, O. Krídlo, B. Macek, L. Pisková, On heterogeneous formal contexts, Fuzzy Sets Syst. 234 (2014) 22–33. [4] R. Bˇelohlávek, Reduction and a simple proof of characterization of fuzzy concept lattices, Fundam. Inform. 21 (2001) 1001–1010. [5] R. Bˇelohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Publishers, New York, 2002. [6] R. Bˇelohlávek, Concept lattices and order in fuzzy logic, Ann. Pure Appl. Logic 128 (2004) 277–298. [7] R. Bˇelohlávek, Sup-t-norm and inf-residuum are one type of relational product: unifying framework and consequences, Fuzzy Sets Syst. 197 (2012) 45–58. [8] R. Bˇelohlávek, Ordinally equivalent data: a measurement-theoretic look at formal concept analysis of fuzzy attributes, Int. J. Approx. Reason. 54 (9) (2013) 1496–1506. [9] R. Bˇelohlávek, V. Vychodil, Reducing the size of fuzzy concept lattices by hedges, in: FUZZ-IEEE 2005, The IEEE International Conference on Fuzzy Systems, Reno (Nevada, USA), 2005, pp. 663–668. [10] R. Bˇelohlávek, V. Vychodil, Fuzzy concept lattices constrained by hedges, J. Adv. Comput. Intell. Intell. Inform. 11 (6) (2007) 536–545. [11] R. Bˇelohlávek, V. Vychodil, Formal concept analysis and linguistic hedges, Int. J. Gen. Syst. 41 (2012) 503–532. [12] R. Bˇelohlávek, B. de Baets, J. Koneˇcný, Granularity of attributes in formal concept analysis, Inf. Sci. 260 (2014) 149–170. [13] A. Burusco, R. Fuentes-Gonzalez, The study of L-fuzzy concept lattice, Mathw. Soft Comput. 3 (1994) 209–218. [14] P. Butka, J. Pócs, J. Pócsová, On equivalence of conceptual scaling and generalized one-sided concept lattices, Inf. Sci. 259 (2014) 57–70. [15] P. Butka, J. Pócs, J. Pócsová, Representation of fuzzy concept lattices in the framework of classical FCA, J. Appl. Math. 2013 (2013) 236725, 7 pages. [16] P. Butka, J. Pócs, Generalization of one-sided concept lattices, Comput. Inform. 32 (2) (2013) 355–370. [17] I.P. Cabrera, P. Cordero, G. Gutiérez, J. Martinez, M. Ojeda-Aciego, On residuation in multilattices: filters, congruences, and homomorphisms, Fuzzy Sets Syst. 234 (2014) 1–21. [18] C. Carpineto, G. Romano, Concept Data Analysis Theory and Applications, J. Wiley, 2004. [19] M.E. Cornejo, J. Medina, E. Ramírez, A comparative study of adjoint triples, Fuzzy Sets Syst. 211 (2013) 1–14. [20] B.A. Davey, H.A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 2002. [21] J.C. Díaz-Moreno, J. Medina, M. Ojeda-Aciego, On basic conditions to generate multi-adjoint concept lattices via Galois connections, Int. J. Gen. Syst. 43 (2) (2014) 149–161. [22] F. Esteva, L. Godo, C. Noguera, A logical approach to fuzzy truth hedges, Inf. Sci. 232 (2013) 366–385. [23] B. Ganter, R. Wille, Formal Concept Analysis Mathematical Foundation, Springer Verlag, 1999. [24] P. Hájek, On very true, Fuzzy Sets Syst. 124 (2001) 329–333. [25] J. Koneˇcný, Isotone fuzzy Galois connections with hedges, Inf. Sci. 181 (2011) 1804–1817. [26] J. Koneˇcný, J. Medina, M. Ojeda-Aciego, Multi-adjoint concept lattices with heterogeneous conjunctors and hedges, Ann. Math. Artif. Intell. 72 (1) (2014) 73–89. [27] J. Koneˇcný, P. Osiˇcka, Triadic concept lattices in the framework of aggregation structures, Inf. Sci. 279 (2014) 512–527. [28] S. Krajˇci, A generalized concept lattice, Log. J. IGPL 13 (2005) 543–550. [29] S. Krajˇci, Every concept lattice with hedges is isomorphic to some generalized concept lattice, in: V. Snášel, R. Bˇelohlávek (Eds.), Proceedings of the 3rd International Conference on Concept Lattices and Their Applications, 2005, pp. 1–9.

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