Characteristics of three-way concept lattices and three-way rough concept lattices

Abstract In classical formal concept (Wille’s concept) analysis, a basic theorem about concept lattices is that every concept lattice is a complete lattice and conversely, every complete lattice is isomorphic to a concept lattice. Three-way concept analysis is an extended theory of formal concept analysis. Similarly, three-way concept lattices and three-way rough lattices are also complete lattices. However, unlike the classical case, not every complete lattice arises as a three-way concept lattice (or as a three-way rough concept lattice). In this paper, we focus on characterising those complete lattices which can be represented by three-way concept lattices. In order to achieve this, we first discuss some properties of special elements such as atoms and irreducible elements, and complements of three-way concept lattices. Then we give our main theorem by displaying conditions under which any complete lattice can be realised as a three-way concept lattice. Similar results are discussed and obtained for three-way rough concept lattices.

[1]  Wei-Zhi Wu,et al.  Three-way concept learning based on cognitive operators: An information fusion viewpoint , 2017, Int. J. Approx. Reason..

[2]  Ling Wei,et al.  The connections between three-way and classical concept lattices , 2016, Knowl. Based Syst..

[3]  Yuhua Qian,et al.  Three-way cognitive concept learning via multi-granularity , 2017, Inf. Sci..

[4]  Ling Wei,et al.  The attribute reductions of three-way concept lattices , 2016, Knowledge-Based Systems.

[5]  James F. Peters,et al.  Proximal three-way decisions: Theory and applications in social networks , 2016, Knowl. Based Syst..

[6]  Bernhard Ganter,et al.  Formal Concept Analysis: Mathematical Foundations , 1998 .

[7]  Jesús Medina,et al.  Multi-adjoint property-oriented and object-oriented concept lattices , 2012, Inf. Sci..

[8]  Ling Wei,et al.  A Novel Concept Acquisition Approach Based on Formal Contexts , 2014, TheScientificWorldJournal.

[9]  Rudolf Wille,et al.  Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts , 2009, ICFCA.

[10]  Guoyin Wang,et al.  Approximate concept construction with three-way decisions and attribute reduction in incomplete contexts , 2016, Knowl. Based Syst..

[11]  Yiyu Yao,et al.  An Outline of a Theory of Three-Way Decisions , 2012, RSCTC.

[12]  Xiao Zhang,et al.  Rule-preserved object compression in formal decision contexts using concept lattices , 2014, Knowl. Based Syst..

[13]  Yee Leung,et al.  Attribute reductions in object-oriented concept lattices , 2014, Int. J. Mach. Learn. Cybern..

[14]  J.M. Rodriguez-Jimenez,et al.  Negative Attributes and Implications in Formal Concept Analysis , 2014, ITQM.

[15]  Bao Qing Hu,et al.  Three-way decisions space and three-way decisions , 2014, Inf. Sci..

[16]  Sang-Eon Han,et al.  Lattice-theoretic contexts and their concept lattices via Galois ideals , 2016, Inf. Sci..

[17]  Yiyu Yao,et al.  Three-way decisions with probabilistic rough sets , 2010, Inf. Sci..

[18]  Yiyu Yao,et al.  The superiority of three-way decisions in probabilistic rough set models , 2011, Inf. Sci..

[19]  Ivo Düntsch,et al.  Modal-style operators in qualitative data analysis , 2002, 2002 IEEE International Conference on Data Mining, 2002. Proceedings..

[20]  Wei Liu,et al.  Computing the set of concepts through the composition and decomposition of formal contexts , 2012, 2012 International Conference on Machine Learning and Cybernetics.

[21]  Yuhua Qian,et al.  Concept learning via granular computing: A cognitive viewpoint , 2014, Information Sciences.

[22]  Decui Liang,et al.  Deriving three-way decisions from intuitionistic fuzzy decision-theoretic rough sets , 2015, Inf. Sci..

[23]  Ling Wei,et al.  Constructing three-way concept lattices based on apposition and subposition of formal contexts , 2017, Knowl. Based Syst..

[24]  Sebastian Rudolph,et al.  Concept lattices with negative information: A characterization theorem , 2016, Inf. Sci..

[25]  Yiyu Yao,et al.  Interval sets and three-way concept analysis in incomplete contexts , 2016, International Journal of Machine Learning and Cybernetics.

[26]  Rokia Missaoui,et al.  INCREMENTAL CONCEPT FORMATION ALGORITHMS BASED ON GALOIS (CONCEPT) LATTICES , 1995, Comput. Intell..

[27]  Jerzy W. Grzymala-Busse,et al.  Rough Sets , 1995, Commun. ACM.

[28]  Radim Bělohlávek,et al.  Fuzzy galois connections and fuzzy concept lattices: from binary relations to conceptual structures , 2000 .

[29]  Lingyun Yang,et al.  On Rough Concept Lattices , 2010, ISDT.

[30]  Xia Wang,et al.  Relations of attribute reduction between object and property oriented concept lattices , 2008, Knowl. Based Syst..

[31]  Jiye Liang,et al.  International Journal of Approximate Reasoning Multigranulation Decision-theoretic Rough Sets , 2022 .

[32]  Qingguo Li,et al.  On the order-theoretic properties of lower concept formula systems , 2014, Soft Comput..

[33]  Yiyu Yao,et al.  Three-Way Decisions and Cognitive Computing , 2016, Cognitive Computation.

[34]  Ming-Wen Shao,et al.  Knowledge reduction in formal fuzzy contexts , 2015, Knowl. Based Syst..

[35]  Qing Wan,et al.  Approximate concepts acquisition based on formal contexts , 2015, Knowl. Based Syst..

[36]  Yiyu Yao,et al.  Three-Way Formal Concept Analysis , 2014, RSKT.

[37]  Bao Qing Hu,et al.  Three-way decision spaces based on partially ordered sets and three-way decisions based on hesitant fuzzy sets , 2016, Knowl. Based Syst..

[38]  Yiyu Yao,et al.  Concept lattices in rough set theory , 2004, IEEE Annual Meeting of the Fuzzy Information, 2004. Processing NAFIPS '04..

[39]  Maokang Luo,et al.  Rough concept lattices and domains , 2009, Ann. Pure Appl. Log..

[40]  Decui Liang,et al.  A novel three-way decision model based on incomplete information system , 2016, Knowl. Based Syst..

[41]  Ling Wei,et al.  The three-way object oriented concept lattice and the three-way property oriented concept lattice , 2015, 2015 International Conference on Machine Learning and Cybernetics (ICMLC).

[42]  Qingguo Li,et al.  Power contexts and their concept lattices , 2011, Discret. Math..

[43]  Yiyu Yao,et al.  A Comparative Study of Formal Concept Analysis and Rough Set Theory in Data Analysis , 2004, Rough Sets and Current Trends in Computing.

[44]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .