1 Introduction
Among several theoretical tools applied in data mining, an important role is played by aggregation functions. Recall, for example, the application of Choquet and Sugeno integrals and other aggregation functions discussed in STKPL . Note that aggregation functions were originally introduced to act on real intervals, for details we refer the reader to the comprehensive monographs BPC ; Grabisch et al 2009 and KMP .
However, recently the aggregation on posets, and in particular on lattices, has became a rapidly growing topic, especially due to applications in information sciences, see e.g. Couceiro , Grabisch et al 2009 etc. This trend was stressed at the international conference ABLAT (Aggregation on Bounded LATtices) held in 2014 in Trabzon, Turkey, among others. Aggregation on lattices exploiting several algebraic results is often bringing new lights also into the standard realvalued aggregation techniques. For example, though the Sugeno integral was introduced in 1974 Sug74 , and discussed in many papers and monographs, only recently, based on algebraic look, it was shown that it is, in fact, an aggregation function which preserves congruences HMP1 ; HMP3 . This allows to apply the Sugeno integral consistently also when we change a numerical scale into linguistic scale, for example. Also quite recently, the essential progress with respect to understanding how aggregation functions can be generated has been achieved, see HMP2 ; HP1 ; HP2 . For more details on recent applications of aggregation functions we refer the reader to T1 ; T2 ; Kara2 ; T4 ; Kara1 ; Lei ; MSY ; MY ; MZ1 ; Ze1 ; MZ2 .
Another useful method used in data mining during last decades is Formal Concept Analysis (FCA, in short) GW . It is a theory where data are analysed by means of conceptual structures among data sets. Mathematically, FCA is based on the notion of a formal context which is represented by two sets, objects and attributes, and by a binary relation between the set of objects and the set of attributes representing the relationship between them. As a result of the process, we obtain socalled formal concepts which correspond to the maximal rectangles in the data set. The set of formal concepts has a structure of a complete lattice (called concept lattice) consisting of all conceptual abstractions (concepts) combining subsets of objects with subsets of shared attributes. FCA has been proved to be an effective tool in many areas of science, besides decision making it has been extensively applied to fields such as knowledge discovery, information retrieval, software engineering etc.
Classical FCA method is used for a binary case where we can sharply decide whether or not a given object has a given attribute. On the other hand, there are many natural practical examples where this most simple case is not appropriate. Namely, in many concrete situations the relation between objects and attributes is not crisp, and then manyvalued or fuzzy description of objectattribute model is more convenient. Therefore, when dealing with imprecise data, uncertainty or even when the information is not complete, this more general setting of FCA method has become an important research topic in the recent years. For more details we recommend the reader the papers AL ; AK ; B01 ; BuPPINS ; MedN3 ; HP3 ; Kardos ; Kr1 ; MedN2 ; MedN1 ; MO ; Med09 ; Pocs ; P3 .
Concept lattices can be viewed from another important equivalent way. Namely, they correspond to Galoisclosed sets with respect to a Galois connection, both in monotone as well as in antitone setting, induced by the incidence relation between the sets of objects and attributes. Remark that Galois connections play a fundamental role in mathematics because of their universality. To be more specific, Galois connections represent a structurepreserving passage between two worlds, the one living on the object side, the second on the attribute side. Consequently, they are inherent with respect to human thinking in a sense that they allow to closely connect certain quite different worlds of hierarchical structures. Ordertheoretically, Galois connections consist of two orderpreserving maps whose composition yields two operators on the respective structures, one closure operator, and the second being a kernel operator. In other words, the two hierarchies living in two different worlds can be transported to each other. Such an adjoint situation has an advantage that the knowledge about one of the worlds can be used to gain the information in the second one. Remind that historically, the classical Galois theory has been used for solving the problem of solvability of algebraic equations. Besides this one can find many other applications of Galoiscorrespondences in almost all branches of mathematics and applied science. As an example, recall the link between conjunctions and implications in fuzzy logics Haj . For more details on Galois connections we recommend the survey paper by Erné et al. Erne .
By authors’ knowledge, so far no essential connection between the theory of aggregation functions and FCA has been developed. The purpose of this paper is to fill this gap by building a bridge between these two theories, one of them living in the world of data fusion, the second one in the area of data mining. Our aim is to show how Galois connections can be used to describe important classes of aggregation functions. Note that the majority of aggregation functions exploited in applications are considered to preserve suprema (e.g., for maximization problems on real intervals) or infima (e.g., for minimization problems). Moreover, on real intervals, simultaneous preservation of suprema and infima by an aggregation function means its continuity. In our paper, we focus on the case of suppreserving aggregation functions only, while the related results for infpreserving aggregation functions can be obtained by duality. As shown below, our method gives an elegant and complete description of this class.
2 Preliminaries
We assume that the reader is familiar with the basic notions and terminology of partially ordered sets, especially lattice theory, cf. Blyth or Gratzer .
The direct product of an indexed system of lattices is defined in the usual way; we apply the notation . If for all , the symbol denotes the direct power of a lattice or we use provided . The elements of the direct product will be denoted by the bold symbols and for , is the th component of in the lattice . Recall that the direct product of lattices forms a complete lattice if and only if all of them are complete lattices.
Given a partially ordered set , a closure operator on is a selfmap which is monotone, extensive and idempotent. More precisely, this means the following conditions for all :

provided


.
The notion of an interior operator is defined dually, i.e., an interior operator on is a mapping which is monotone, intensive (i.e., for all ) and idempotent.
In the case of complete lattices, the notions of closure and interior operators are closely related to that of closure systems and interior systems, respectively. Given a complete lattice , a closure system on is a subset closed under arbitrary infima, i.e.,
Dually, an interior system on is a subset closed under arbitrary suprema, i.e.,
Let us remark that as and holds in any complete lattice, the top element belongs to every closure system and analogously, every interior system contains the bottom element .
It is the wellknown fact that every closure operator gives rise to a closure system and vice versa. For a closure system on , one can define by . Conversely, given a closure operator on , the set of fixed points forms a closure system on . In this case and .
A similar correspondence holds between interior operators and interior systems, in this case represents the interior operator corresponding to an interior system .
Further, we recall the definition and basic properties of monotone Galois connections. The results presented in this section can be found in several sources, however in a noncompact form, c.f. Blyth or Erne . In order to make the paper as selfcontained as possible, we provide the necessary results in a modified comprehensive form together with their proofs.
Let , be two posets. We say that a pair of mappings , and forms a monotone Galois connection if for all and it holds
(1) 
Then is called the lower adjoint of , while is referred to as the upper adjoint of . Note that given a mapping , there is at most one upper adjoint satisfying (1). To see this, consider two such mappings and . From (1) we immediately infer if and only if , i.e., the sets of lower bounds of and coincide, implying that . The uniqueness of the lower adjoint (if it exists) corresponding to a given mapping can be shown similarly.
Let be a monotone Galois connection between and . Then and are both monotone and they satisfy and for all and . Indeed, from (1) we easily obtain that implies . Consequently, applying (1) again, the inequality yields , i.e., is monotone.
Consequently, we obtain the following important property:
(2) 
To see this, using (1), from we obtain . On the other hand, and the monotonocity of yield .
Let be a complete lattice. For an element , denotes the principal ideal generated by , i.e., . Dually, denotes the principal filter generated by . A subset is called hereditary, or a downset, if for every and with we have . The concept of an upset is defined dually.
Proposition 2.1.
Let and be complete lattices and be a mapping. The following conditions are equivalent.

is preserving.

The inverse image is a downset for every .

There exists an upper adjoint mapping of .
Proof.
Let and . As is preserving, it is monotone. Thus, for we obtain , which yields that , i.e., the set is hereditary. Further, put . As for all , we obtain
showing that . Since is a downset and is its greatest element, we have .
For put where is such that . Then obviously
Using the dual arguments, one can similarly prove the following.
Proposition 2.2.
Let and be complete lattices and be a mapping. Then the following conditions are equivalent:

is preserving.

The inverse image is an upset for every .

There exists the lower adjoint mapping of .
Hence, as a conclusion of the above propositions we obtain that given a preserving mapping , there is the unique preserving mapping such that and form a monotone Galois connection.
For a mapping , let denotes its range. The following important assertion provides an inner description of monotone Galois connections.
Proposition 2.3.
Let and be complete lattices and , be two mappings between them. If the pair forms a monotone Galois connection then

the range is an interior system on ,

the range is a closure system on ,

and are isomorphic posets.
Conversely, let be a closure system on with the corresponding closure operator , an interior system on with the corresponding interior operator and an isomorphism. Then the mappings , form a monotone Galois connection.
Proof.
Obviously, is an interior system on since is preserving. Similarly, is a closure system on . Further, as and satisfy (2), we have for all as well as for all . Thus, we obtain that restricted to and restricted to are mutually inverse. As both these mappings are monotone, and are isomorphic posets.
Conversely, assume that and are isomorphic. Since and , from the basic properties of closure and interior operators we obtain
which is equivalent to
∎
In the sequel, if a pair of isomorphic closureinterior systems is considered, it is implicitly assumed that some isomorphism between them is also present. Moreover, we will consider (ary) aggregation functions , i.e., functions which are characterized by the monotonicity and boundary conditions. Hence is an aggregation function whenever if , and . In particular, each homomorphism is an aggregation function.
3 Suppreserving and infpreserving aggregation functions
Recall that any aggregation function on a complete lattice fulfills the boundary conditions
where and denote the bottom and the top element of the lattice . In order to apply Proposition 2.3, these conditions give the following basic characterization:
Lemma 3.1.
Let be a complete lattice and be an aggregation function. Then is preserving if and only if for all , where is an isomorphism between a closure system and an interior system such that , and is the closure operator corresponding to .
Similarly, is preserving if and only if for all , where is an isomorphism between an interior system and a closure system such that , and is the interior operator corresponding to .
Proof.
Assume that is a preserving aggregation function. According to Proposition 2.3, is determined by isomorphic closure system and interior system via some isomorphism . As and (every closure system contains the top element), we obtain
and it follows that .
Conversely, assume that . Obviously, is preserving. Since the closure equals to , the least element of , we obtain
Note that the last equality follows from the fact that maps the least element of into the least element of , which is . Dually, for the greatest element of we obtain
since maps the greatest element of into the greatest element .
The assertion concerning the preserving aggregation functions can be proved analogously. ∎
Although the previous lemma provides the basic inner characterization of the preserving as well as the preserving mappings, closure and interior systems on a direct power need not be so transparent. In what follows we try to find a similar characterization, however with respect to possibly more simple factors.
Obviously, any preserving aggregation function is decomposable in the following way
(3) 
where for all is preserving. In this case, for each the function is the lower adjoint of , where is the upper adjoint of and denotes the th projection.
Conversely, given any system of of preserving mappings, the formula (3) determines a preserving function . In addition, being an aggregation function, the functions of the system have to satisfy . Hence, there is a onetoone correspondence between the family of all ary preserving aggregation functions on a complete lattice and the family whose elements consist of pairs of closureinterior system pairs on , satisfying , where is a greatest element in .
Notice that a similar characterization can be also applied for preserving aggregation functions.
Example 3.2.
Consider the sixelement lattice whose Hasse diagram is depicted in Fig. 1.
In order to generate a binary preserving aggregation function, consider two pairs of isomorphic closureinterior systems and on . In this case and . Both systems are indicated by filled circles, where is depicted on the left side and on the right side of Fig. 1. The systems and are denoted by slashed circles, particularly and . Let us note that for the pair , the corresponding isomorphism between these two systems is unique. However, considering the pair , the corresponding isomorphism is indicated by the arrow, i.e., . Then we necessarily have , and the top and the bottom elements are mapped into their respective counterparts. As the top elements of and satisfy , the considered two pairs of isomorphic closureinterior systems induce a preserving aggregation function , the values of which are given in Table 1.
For example, the value is calculated in the following way. First, the closures and are determined. In the first case, is the smallest element in which is above and, obviously, it is itself. In the second case, is the smallest element in which is above , which is . Finally,
Further, we describe a decomposition in the case when an underlying complete lattice is a direct product of an indexed system of complete lattices.
Theorem 3.3.
Let and be indexed families of complete lattices. Then there is a preserving mapping if and only if there is a system of preserving mappings such that
(4) 
Proof.
First, assume that there is the above system of preserving mappings. Let be a family of elements and let be defined by (4). Then for each
where the last equality follows from the fact that for each and the mapping is preserving. However, due to basic properties of the supremum operation, we obtain
Conversely, assume that is a preserving mapping. For and define as , where is given by if and otherwise. It is easily seen that every is preserving. Further, for all we obtain
∎
Corollary 3.4.
Let be a preserving mapping defined by (4). Then its preserving adjoint is given by
(5) 
where for each and , is the upper adjoint of .
Proof.
In particular cases, the last two results can be significantly strengthened. For example, consider an associative symmetric aggregation function with a neutral element , i.e., is a triangular norm KMP of the product lattice . Then, due to BM , is preserving (preserving) if and only if is a product of classical preserving (preserving) tnorms, i.e., there are triangular norms , , which are preserving (preserving), and .
Example 3.5.
Consider the lattice from Example 3.2. It can be easily seen that , where and denote the threeelement and twoelement chains respectively, with the usual order. We present the decomposition of the preserving mapping determined by the closureinterior system from Example 3.2. According to Theorem 3.3, there is a system of preserving mappings between the particular direct factors. The resulting decomposition via closureinterior systems is given in Fig. 2.
In the finite case, every lattice is a direct product of directly indecomposable lattices. However, the structure of directly indecomposable lattices may be difficult, e.g., it is evident that every lattice with a prime number of elements is directly indecomposable. Hence, in order to obtain a representation of preserving aggregation functions with respect to possibly simpler lattices, we turn our attention to the socalled subdirect products, cf. Gratzer .
Recall that a lattice is a subdirect product of an indexed family of lattices if

is a sublattice of ,

for each , i.e., each coordinate projection maps onto the corresponding factor.
In such a case we shall write . A subdirect representation of a lattice is any embedding whose image is a subdirect product. A lattice is said to be subdirectly irreducible if and all subdirect representations of are trivial, i.e., if then necessarily for some index . In other words, a lattice is subdirectly irreducible if it is not subdirectly representable by “simpler” lattices. Note that subdirectly irreducible lattices play a similar role with respect to subdirect products of lattices as primes with respect to multiplication of integers.
Recall the wellknown fact that every lattice is isomorphic to a subdirect product of subdirectly irreducible lattices, which are homomorphic images of . From this point of view, the subdirectly irreducible lattices can be considered to have a simpler structure than the former lattice.
As subdirectly irreducible factors of a subdirect representation can be found within the homomorphic images, it follows that every finite lattice is isomorphic to a subdirect product of finite numbers of finite subdirectly irreducible lattices.
For the sake of simplicity, when dealing with a subdirect representation, we limit our attention to the finite case.
First, observe the following two simple but important facts. If , and , are monotone Galois connections, then the composition , is a monotone Galois connection as well. Indeed, for all and we have
Further, given a finite lattice and a sublattice , let and be the corresponding closure and interior operators, respectively. Note that is both closure and interior system on , since is a sublattice. Then the pair , considering as a mapping and being the identity inclusion of into , forms a monotone Galois connection between and its sublattice . This follows from the defining properties of the closure operators, in fact for any and we obtain
Similarly, the pair , , forms a monotone Galois connection between and , since for all and
Lemma 3.6.
Let be a finite lattice and be a sublattice. Then any preserving function is given by
(6) 
where is a preserving mapping.
In this case, the upper adjoint of is given by
(7) 
where is the upper adjoint of .
Proof.
Obviously, if and are given by (6) and (7) respectively, then
Since the pairs of mappings , and form monotone Galois connections, their composition pair forms a monotone Galois connection as well.
Further, assume that is a preserving mapping. Then we can extend to a preserving mapping with the domain as follows:
The mapping is a composition of preserving mappings since is the lower adjoint of , and thus it is preserving. Moreover, for all we have and , which yields
Finally, this shows that can be expressed by (6). ∎
Theorem 3.7.
Let be a finite lattice, be a finite family of finite lattices such that and let , be the corresponding closure and interior operators on . Then the following conditions are equivalent:

is a preserving aggregation function.

For each there is a system of preserving mappings satisfying . Moreover, for all , and for each , are given by (4).
Example 3.8.
Consider the lattice from Example 3.2. It is the wellknown fact that the only subdirectly irreducible distributive lattice is isomorphic to the two element chain . As is distributive, it has a subdirect representation into the direct power of two element chains. Fig. 3 shows such a subdirect representation of in .
With respect to this representation, any preserving mapping is determined by a system of closureinterior systems on . We present such a system, corresponding to the preserving mapping determined by the pair from Example 3.2. Particularly, as is subdirectly represented in , we have pairs of closureinterior systems. They are depicted in Fig. 4. The system in the th row and in the th column corresponds to the mapping between the th factor and the th factor within the subdirect representation of .
Let us note, that in a similar way the subdirect decomposition of the mapping corresponding to pair can be obtained. Hence the whole aggregation function from Example 3.2 can be characterized as a matrix consisting of isomorphic closureinterior pairs on .
4 Relationship to the FCA based biclustering methods
In this section we briefly discuss possible applications of the results from the previous section within certain biclustering fuzzy FCAbased methods.
As it is common in cluster analysis, an
data matrix is given by objects , attributes and entries . The primary aim of bicluster analysis is to identify subgroups of objects which are as similar as possible to each other with respect to some subset of attributes , and different as much as possible to the rest of objects and attributes. Bicluster is then formally defined as a pair .Such a relatively wide definition of biclustering certainly fulfills the classical FCA, cf. Ifca where some link between FCA and biclustering can be found. In this case a data matrix contains only  values and it can be formally seen as an incidence relation between the objects and attributes, i.e., . Then or equivalently is interpreted as “an object has an attribute ”. The biclusters, socalled formal concepts, correspond to the maximal rectangles in the data sets. Given a formal concept , is a subset of all attributes shared by all objects of , while is a subset of objects sharing all attributes of . This “sharing” of attributes represents the essence of the similarity, mentioned in the definition of biclustering. It turns out that formal concepts can be defined via concept forming operators, acting between the power set of objects and the power set of attributes. These operators are induced by the relation
The formal concepts are precisely the fixed points of the operators, i.e., pairs fulfilling and . Such defined concept forming operators form an antitone Galois connection between and , cf. GW .
Having this in mind, several biclusteringlike fuzzy approaches to FCA were proposed, either with the help of antitone or monotone Galois connections B01 ; Kr1 ; MedN3 ; MedN2 ; MedN1 .
To mention an example of such fuzzy concept forming operators, consider the socalled monotone Galois connections, introduced in GePo . In this case, is a complete commutative residuated lattice and is an relation. The concept forming operators and are given by
These concept forming operators are typically involved when some type of fuzzy logic, with as the truth value structure, is used for evaluation membership degrees of particular attributes. The operation is a fuzzy counterpart of the classical logical conjunction, while stands for a fuzzy implication. From a fuzzy logical point of view, the concept forming operators admit the following interpretation: represents the truth degree of the proposition “there exists an object having an attribute ”, where the fuzzy subset corresponds to , and represents the truth degree of the proposition “for all attributes , provided an object is in relation with ”, taking the role of .
In each residuated lattice, the two fuzzy connectives and are related by adjoint property
Hence each determines a preserving mapping , for all , with the upper adjoint for all . Observe that concept forming operators and are defined by (4) and (5), respectively.
The previous considerations allow to define a slightly modified, but more general approach, not connected with any fuzzy logic framework. Let be a fixed complete lattice, be a set of objects and be a set of attributes characterizing particular objects. As a basic input, consider a data table in the form of a manyvalued binary relation , where represents some set of possible alternatives for a characterization of objects from by particular attributes from . To induce concept forming operators from such input data, we use a mapping from the set into the set of preserving mappings, where denotes a mapping associated to an element . Then applying Theorem 3.3 and its corollary we obtain the mappings and defined by
(8) 
for all , and
(9) 
for all , form a monotone Galois connection between them. The biclusters, or fuzzy formal concepts, are defined as the fixed points of these operators, i.e. is a fuzzy formal concept if and .
From Proposition 2.3 we obtain that the set of all concepts, partially ordered by if (or equivalently, ), has a lattice structure. Particularly, this concept lattice is isomorphic to the induced interior system on , which is also isomorphic to the induced closure system on .
Taking into account a natural condition that the top elements of and should form a formal concept, we obtain that for each the composition of and the projection map forms a preserving aggregation function. Hence the basic concept forming operator can also be seen as a system of preserving mappings, where any of them is in some sense determined by the values , varying through the set of all objects . Given a fixed attribute , the mapping assigning to each the value can be seen as some kind of weighted supremum of the values . Consequently, formal concepts can be studied and interpreted within many other theories, where these types of functions play an important role, e.g., in multicriterial decision support. Also the mapping can be understood as some valued possibility measure on fuzzy sets, being a fuzzy analogy of possibility measures. Such view allows to consider about FCAbased clustering methods in the realm of the possibility theory. These different perspectives on the mentioned fuzzy FCAbased biclustering method can be useful in order to incorporate other types of information usually available for the considered data.
5 Conclusion
Formal concepts can be studied and interpreted within many different theories, where these types of clusters play an important role, e.g., in multicriteria decision support, or in possibility theory. In this contribution, we have focused on links between aggregation functions acting on complete lattices and formal concept analysis. In particular, we have elaborated a description of sup (inf) preserving aggregation functions, thus generalizing several particular results known from the literature (such as the structure of sup and infpreserving triangular norms and conorms on product lattices characterized in BM ). We have also discussed possible applications of our results within certain biclustering fuzzy FCAbased methods. We believe that our approach and examples of applications of the general methods of formal concept analysis will expand to several new areas, offering them a powerful tool.
Acknowledgment
The first author was supported by the international project Austrian Science Fund (FWF)Grant Agency of the Czech Republic (GAČR) no. 1534697L; the second author by the Slovak Research and Development Agency under contract APVV140013; the third author by the IGA project of the faculty of Science Palacký University Olomouc no. PrF2016006 and by the Slovak VEGA Grant no. 2/0044/16.
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