Journal of Applied Mathematics

Journal of Applied Mathematics / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 236725 | https://doi.org/10.1155/2013/236725

Peter Butka, Jozef Pócs, Jana Pócsová, "Representation of Fuzzy Concept Lattices in the Framework of Classical FCA", Journal of Applied Mathematics, vol. 2013, Article ID 236725, 7 pages, 2013. https://doi.org/10.1155/2013/236725

Representation of Fuzzy Concept Lattices in the Framework of Classical FCA

Academic Editor: Hak-Keung Lam
Received12 Jul 2013
Revised30 Sep 2013
Accepted10 Oct 2013
Published17 Nov 2013

Abstract

We describe a representation of the fuzzy concept lattices, defined via antitone Galois connections, within the framework of classical Formal Concept Analysis. As it is shown, all needed information is explicitly contained in a given formal fuzzy context and the proposed representation can be obtained without a creation of the corresponding fuzzy concept lattice.

1. Introduction

Formal concept analysis (FCA) is a theory of data analysis for identification of conceptual structures among datasets. The mathematical theory of FCA is based on the notion of concept lattices and it is well developed in the monograph of Ganter and Wille [1]. In this classical approach to FCA the authors provide the crisp case, where an object-attribute model is represented by some binary relation. In practice there are natural examples of object-attribute models for which the relationship between the objects and the attributes is represented by fuzzy relation. Therefore, several attempts to fuzzify FCA have been proposed. There are two kinds of existing approaches to fuzzy FCA based on the structure of the concept lattices. In the first case the concept lattices are fuzzy complete lattices (or complete -lattices) (cf. [2, 3] or [4]). In the second case the concept lattices are ordinary (crisp) complete lattices. From these approaches we mention a work of Bělohlávek [57] based on the logical framework of complete residuated lattices, a work of Georgescu and Popescu to extend this framework to noncommutative logic [810], the approaches of Krajči [11], Popescu [12], and Medina et al. [13, 14], or other works on fuzzy concept lattices [1519]. A nice survey and comparison of some existing approaches to fuzzy concept lattices is available in [20].

Recently, a generalization of Popescu's approach [12] for creating crisp fuzzy concept lattices was introduced (cf. [2123]) for the so-called one-sided concept lattices. This method is in some manner the most general one and it covers most of the mentioned approaches based on the antitone Galois connections. The main feature of this approach is that it is not fixed to any logical framework and it provides a possibility to create concept lattices also in cases, where particular objects and attributes have assigned different complete lattices representing their truth value structures. Moreover this approach generates all possible antitone Galois connections between the products of complete lattices. According to these facts, in this paper we will use the definition of fuzzy concept lattice as it was proposed in [21].

The main goal of this paper is to describe a representation of the fuzzy concept lattices (represented as crisp complete lattices) in the framework of classical concept lattices. It is a well-known fact that every complete lattice is isomorphic to some concept lattice. Hence, for a given fuzzy formal context, it is possible to create the fuzzy concept lattice first and then find a representation of this fuzzy concept lattice as a classical concept lattice. However, our goal is to describe a representation of the fuzzy concept lattices in the classical FCA framework without any previous creation of the fuzzy concept lattices. We only use the information (knowledge), which is explicitly contained in the given formal fuzzy context.

Since the theory of concept lattices is closely related to antitone Galois connections and closure systems, we give a brief overview of these notions in the preliminary section. Further we describe an approach of creating fuzzy concept lattices as it was defined in [21].

In Section 3 we prove our main theorem, that is, an isomorphism between the fuzzy and the classical concept lattices is described. The proof of this theorem involves the principal ideal representation of complete lattices. At the end of this section we also provide an illustrative example of such representation.

2. Preliminaries

In this section we mention some results concerning the antitone Galois connections and their relationship with the closure systems in complete lattices. Also, we briefly recall the framework of classical FCA as well as the notion of fuzzy concept lattices as it was presented in [21]. In the sequel we will assume that the reader is familiar with the basic notions of lattice theory (cf. [24]).

First we recall the definition of the antitone (contravariant) Galois connections (see [25] or [1]).

Let and be ordered sets and let be maps between these ordered sets. Such a pair of mappings is called an antitone Galois connection between the ordered sets if(a)   implies , (b)   implies , (c)   and .

The two maps are also called dually adjoint to each other. We note that and that the conditions (a), (b), and (c) are equivalent to the following one:(d)   if and only if .

In what follows we will denote by the set of all antitone Galois connections between the partially ordered sets and . The class of all complete lattices will be denoted by .

The antitone Galois connections between complete lattices are closely related to the notion of closure operator and closure system. Let be a complete lattice. By a closure operator in we understand a mapping satisfying(a) for all , (b) for , (c) for all (i.e., is idempotent).

A subset of a complete lattice is called a closure system in if is closed under arbitrary meets. We note that this condition guarantees that is a complete lattice, in which the infima are the same as in , but the suprema in may not coincide with those from . It is well known that closure systems and closure operators are in one-to-one correspondence; that is, the closure operator associated with a closure system defines the closure of an element as the least closed element containing and the closure system associated with a closure operator is the family of its fixed points ().

The following result (see [25]) relates the relationship between the antitone Galois connections and dually isomorphic closure systems of the complete lattices.

Proposition 1. Let and be an antitone Galois connection between and . Then the mapping is a closure operator in , and similarly, is a closure operator in . Moreover, the corresponding closure systems are dually isomorphic.
Conversely, suppose that and are closure systems in and , respectively, and is a dual isomorphism between the complete lattices and . Then the pair , where , are closure operators corresponding to and to , forms an antitone Galois connection between and .

We also recall another useful characterization of the antitone Galois connections between complete lattices (see [1]).

Proposition 2. A map between complete lattices and has a dual adjoint if and only if holds for any subset of .
Note that in this case the dual adjoint is uniquely determined by

The properties of the antitone Galois connections allow constructing complete lattices (Galois lattices). Formally, let and be an antitone Galois connection between and . Denote by a subset of consisting of all pairs with and . Define a partial order on as follows:

Proposition 3. Let and be an antitone Galois connection between and . Then forms a complete lattice, where for each family of elements from .

Now we briefly recall the basic notions of FCA [1].

Let be a formal context, that is, and . There is a pair of mappings and , which forms an antitone Galois connection between the power sets and The corresponding concept lattice is denoted by (cf. Proposition 3).

Next, we describe the approach proposed in [21]. We start with the definition of a formal fuzzy context.

A 6-tuple is called a formal fuzzy context if (i) ( is the set of objects and is the set of attributes),(ii), (recall that denotes the class of all complete lattices, and thus for , represents a complete lattice with possible truth values of the object and similarly for ),(iii), where for each , .

Further, define as follows:

Similarly we put as follows:

The following theorem shows the relation between the mappings defined above and the antitone Galois connections between the direct products of the complete lattices. The proof of this theorem can be found in [21].

Proposition 4. Let be a formal fuzzy context. Then the pair forms an antitone Galois connection between and .
Conversely, let be an antitone Galois connection between and . Then there exists a formal fuzzy context , such that and .

According to this proposition and Proposition 3, the lattice corresponding to the formal fuzzy context will be denoted by .

Let us note that the second part of Proposition 3 allows a representation of any fuzzy concept lattice (created by antitone Galois connection) as for a suitable formal fuzzy context .

As an example we show such representation of the approach based on residuated lattices [5, 7].

Let be a complete residuated lattice and be an arbitrary element. Denote by a mapping with . Since for all , we obtain that the pair forms an antitone Galois connection between and . Hence, if is -context, then one can easily obtain formal fuzzy context in our sense by choosing and .

3. Representation of Fuzzy Concept Lattices in Classical FCA

In this section we describe the theoretical details regarding the representation of fuzzy concept lattices in the framework of classical concept lattices. We prove a theorem which shows that our representation of a fuzzy concept lattice as a classical concept lattice is correct; that is, we show that both lattices are isomorphic. At the end of this section we provide an illustrative example of such representation.

Let be any complete lattice. For an element denote by the principal ideal generated by the element .

Let be a formal fuzzy context. In the sequel we will suppose that and form pairwise disjoint family of sets. For each and each define a binary relation by and let

Obviously, the triple forms a classical formal context.

Lemma 5. Let , be arbitrary elements and and . Then

Proof. Obviously, for all if and only if for all . According to Proposition 2 this is equivalent to .

Let be an object and be an attribute. For a concept we put and . The next lemma shows that each of the subsets forms a principal ideal in . The same is true for .

Lemma 6. Let be a concept. Then for all and for all .

Proof. We prove for all . Let be an object. Since any is lower than or equal to , the inclusion is trivial.
Now we prove the opposite inclusion. Let be an arbitrary element. There is an attribute such that and for all , which is equivalent to for all by (11). Let . From the basic properties of antitone Galois connections and due to Proposition 2 we obtain which yields and consequently for all . Since this holds for all , we obtain .

Lemma 7. Let be a concept. Then

Proof. We prove for all . Let be an arbitrary element. Since , we obtain for all , and for all . Due to Lemma 5 this is equivalent to for all which yields .
Conversely, denote . Then for all , and hence according to Lemma 5   for all , . The pair forms a concept; thus , and we obtain .

Now we can provide our main theorem.

Theorem 8. Let be a formal fuzzy context. Then and this isomorphism is given by the following correspondence:

Proof. Let be the mapping defined in the theorem.
First, we prove that the range of is a subset of , that is, for all , satisfying and .
The following chain of equivalent assertions shows : In the same way one can prove .
Since in any lattice we have if and only if , we obtain that the mapping satisfies for all . Let us remark that this condition ensures that the mapping is injective.
Finally, we show that the mapping is surjective. Let be a concept. Define and by According to Lemma 6   for all and for all , thus Moreover, Lemma 7 shows that for all and for all , which completes the proof.

In practice, one can use the result of this theorem as follows. Let be a given formal fuzzy context. There is given the formal context as we described. The lattice can be fully reconstructed from the concept lattice using the inverse mapping . In this case for all and for all .

At the end of this section, we provide an illustrative example.

Example 9. We will consider the following formal fuzzy context , where , and the mappings , and the system of antitone Galois connections are introduced in Table 1. Note that for more legibility we only indicate the corresponding dual isomorphism of the closure systems, since it uniquely determines the corresponding antitone Galois connection (cf. Proposition 1).
In this case, our representation gives the formal context (see Table 2). From this context we obtain the following concept lattice (see Figure 1).
Finally, we obtain the fuzzy concept lattice (Figure 2) consisting of pairs , such that for and for , where .


ObjectAttribute
\236725.tab.002 \236725.tab.003 \236725.tab.004

236725.tab.005236725.tab.006236725.tab.007236725.tab.008

236725.tab.009236725.tab.0010236725.tab.0011236725.tab.0012



1111111111111
1100011111111
1111111010100
1100010011000
1100011010100
1100010010000
1100011010100
1100010010000
1111111111111
1010011111111
1000011010100
1000011010100

4. Conclusion

In this paper we described a representation of the fuzzy concept lattices in the framework of the classical FCA. This representation transforms a fuzzy formal context into a binary formal context. As it was shown, this transformation maintains all the information given by the lattice structure of a concept lattice, since the corresponding concept lattices are isomorphic. Consequently, the well developed theory of classical FCA can be used for studying the fuzzy concept lattices or an arbitrary algorithm for classical concept lattices can be used for the creation of the fuzzy concept lattices.

Acknowledgments

The authors would like to thank the anonymous reviewers for their helpful and constructive comments which helped enhancing this paper. This work was supported by the Slovak Research and Development Agency under contracts APVV-0208-10, APVV-0035-10, and APVV-0482-11, by the Slovak VEGA Grants 1/1147/12, 1/0729/12, and 1/0497/11, and by the ESF Fund CZ.1.07/2.3.00/30.0041.

References

  1. B. Ganter and R. Wille, Formal Concept Analysis: Mathematical Foundations, Springer, Berlin, Germany, 1999. View at: Publisher Site | MathSciNet
  2. R. Bělohlávek, “Concept lattices and order in fuzzy logic,” Annals of Pure and Applied Logic, vol. 128, no. 1–3, pp. 277–298, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  3. H. Lai and D. Zhang, “Concept lattices of fuzzy contexts: formal concept analysis vs. rough set theory,” International Journal of Approximate Reasoning, vol. 50, no. 5, pp. 695–707, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. L. Shen and D. Zhang, “The concept lattice functors,” International Journal of Approximate Reasoning, vol. 54, no. 1, pp. 166–183, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. R. Bělohlávek, “Lattices generated by binary fuzzy relations,” Tatra Mountains Mathematical Publications, vol. 16, pp. 11–19, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  6. R. Bělohlávek, “Fuzzy Galois connections,” Mathematical Logic Quarterly, vol. 45, no. 4, pp. 497–504, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. R. Bělohlávek, “Lattices of fixed points of fuzzy Galois connections,” Mathematical Logic Quarterly, vol. 47, no. 1, pp. 111–116, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. G. Georgescu and A. Popescu, “Concept lattices and similarity in non-commutative fuzzy logic,” Fundamenta Informaticae, vol. 53, no. 1, pp. 23–54, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  9. G. Georgescu and A. Popescu, “Non-commutative fuzzy Galois connections,” Soft Computing, vol. 7, pp. 458–467, 2003. View at: Google Scholar | Zentralblatt MATH
  10. G. Georgescu and A. Popescu, “Non-dual fuzzy connections,” Archive for Mathematical Logic, vol. 43, no. 8, pp. 1009–1039, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. S. Krajči, “A generalized concept lattice,” Logic Journal of the IGPL, vol. 13, no. 5, pp. 543–550, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. A. Popescu, “A general approach to fuzzy concepts,” Mathematical Logic Quarterly, vol. 50, no. 3, pp. 265–280, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  13. J. Medina, M. Ojeda-Aciego, and J. Ruiz-Calviño, “On multi-adjoint concept lattices: definition and representation theorem,” in Formal Concept Analysis, Lecture Notes in Artificial Intelligence, pp. 197–209, 2007. View at: Google Scholar | Zentralblatt MATH
  14. J. Medina, M. Ojeda-Aciego, and J. Ruiz-Calviño, “Formal concept analysis via multi-adjoint concept lattices,” Fuzzy Sets and Systems, vol. 160, no. 2, pp. 130–144, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  15. Ch. Aswani Kumar and S. Srinivas, “Concept lattice reduction using fuzzy K-Means clustering,” Expert Systems with Applications, vol. 37, no. 3, pp. 2696–2704, 2010. View at: Publisher Site | Google Scholar
  16. S. Elloumi, J. Jaam, A. Hasnah, A. Jaoua, and I. Nafkha, “A multi-level conceptual data reduction approach based on the Lukasiewicz implication,” Information Sciences, vol. 163, no. 4, pp. 253–262, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. L. Li and J. Zhang, “Attribute reduction in fuzzy concept lattices based on the T implication,” Knowledge-Based Systems, vol. 23, pp. 497–503, 2010. View at: Publisher Site | Google Scholar
  18. X. Liu, T. Chai, W. Wang, and W. Liu, “Approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory. I,” Information Sciences, vol. 177, no. 4, pp. 1007–1026, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  19. X. Liu, W. Wang, T. Chai, and W. Liu, “Approaches to the representations and logic operations of fuzzy concepts in the framework of axiomatic fuzzy set theory. II,” Information Sciences, vol. 177, no. 4, pp. 1027–1045, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  20. R. Belohlavek and V. Vychodil, “What is a fuzzy concept lattice?” in Proceedings of the 3rd International Conference on Concept Lattices and Their Applications (CLA '05), pp. 34–45, Olomouc, Czech Republic, 2005. View at: Google Scholar
  21. J. Pócs, “Note on generating fuzzy concept lattices via Galois connections,” Information Sciences, vol. 185, no. 1, pp. 128–136, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  22. J. Pócs, “On possible generalization of fuzzy concept lattices using dually isomorphic retracts,” Information Sciences, vol. 210, pp. 89–98, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  23. P. Butka and J. Pócs, “Generalization of one-sided concept lattices,” Computing and Informatics, vol. 32, no. 2, pp. 355–370, 2013. View at: Google Scholar | MathSciNet
  24. B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, New York, NY, USA, 2nd edition, 2002. View at: MathSciNet
  25. O. Ore, “Galois connexions,” Transactions of the American Mathematical Society, vol. 55, pp. 493–513, 1944. View at: Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2013 Peter Butka et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 Views |  Downloads |  Citations
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.