Describing hierarchy of concept lattice by using matrix

Abstract Concept lattices (also called Galois lattices) are complete ones with the hierarchical order relation of the formal concepts defined by a formal context or Galois connection. In this paper, we present a new of method describing a hierarchy of a finite concept lattice by using a matrix. Given a finite concept lattice L, we introduce Scott topology σ(L) on L and choose an order of a unique minimal base for σ(L). Then, there is a one-to-one correspondence between the finite topological space (L, σ(L)) and a proper square matrix with integral entries; thus we obtain a hierarchy-matrix describing the hierarchy of the concept lattice. We explain how to get the information of the hierarchy from the hierarchy-matrix and discuss the relation between the hierarchy-matrix and the Hasse diagram. Since the hierarchy-matrix allowed us to store the information of hierarchy of the concept lattice, we believe that any software autonomously understand the information of hierarchy of the concepts from the hierarchy-matrix.

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

[2]  Zuping Zhang,et al.  An efficient algorithm for increasing the granularity levels of attributes in formal concept analysis , 2016, Expert Syst. Appl..

[3]  Qingguo Li,et al.  Formal query systems on contexts and a representation of algebraic lattices , 2013, Inf. Sci..

[4]  Gianpiero Cattaneo,et al.  On the connection of hypergraph theory with formal concept analysis and rough set theory , 2016, Inf. Sci..

[5]  Edwin Diday,et al.  Maximal and Stochastic Galois Lattices , 2003, Discret. Appl. Math..

[6]  K. Hofmann,et al.  Continuous Lattices and Domains , 2003 .

[7]  Cherif Chiraz Latiri,et al.  CoMRing: A Framework for Community Detection Based on Multi-relational Querying Exploration , 2016, KES.

[8]  Chen Hao,et al.  Optimal scale selection in dynamic multi-scale decision tables based on sequential three-way decisions , 2017, Inf. Sci..

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

[10]  Jozef Pócs,et al.  On concept reduction based on some graph properties , 2016, Knowl. Based Syst..

[11]  Zhuo Zhang,et al.  Constructing L-fuzzy concept lattices without fuzzy Galois closure operation , 2018, Fuzzy Sets Syst..

[12]  László Kovács,et al.  Generalization of String Transformation Rules using Optimized Concept Lattice Construction Method , 2017 .

[13]  Yiyu Yao,et al.  Rough-set concept analysis: Interpreting RS-definable concepts based on ideas from formal concept analysis , 2016, Inf. Sci..

[14]  Jinkun Chen,et al.  Relations of reduction between covering generalized rough sets and concept lattices , 2015, Inf. Sci..

[15]  Jan Konecny.  On attribute reduction in concept lattices: Methods based on discernibility matrix are outperformed by basic clarification and reduction , 2017, Inf. Sci..

[16]  R. E. Stong,et al.  Finite topological spaces , 1966 .

[17]  Ming-Wen Shao,et al.  Attribute reduction in generalized one-sided formal contexts , 2017, Inf. Sci..

[18]  Sérgio M. Dias,et al.  Knowledge reduction in formal contexts using non-negative matrix factorization , 2015, Math. Comput. Simul..

[19]  Stanislav Krajci,et al.  On heterogeneous formal contexts , 2014, Fuzzy Sets Syst..

[20]  Sérgio M. Dias,et al.  A methodology for analysis of concept lattice reduction , 2017, Inf. Sci..

[21]  Sujing Wang,et al.  An Algorithm Based on Concept-Matrix for Building Concept Lattice with Hasse , 2007, 2007 International Conference on Wireless Communications, Networking and Mobile Computing.

[22]  Yuhua Qian,et al.  Three-way cognitive concept learning via multi-granularity , 2017, 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]  Zhang Wen-xiu,et al.  Attribute reduction theory and approach to concept lattice , 2005 .

[25]  Peter Butka,et al.  Generalization of One-Sided Concept Lattices , 2013, Comput. Informatics.

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

[27]  Anne Berry,et al.  Representing a concept lattice by a graph , 2002, Discret. Appl. Math..

[28]  Dan Meng,et al.  Formal concept analysis based on the topology for attributes of a formal context , 2013, Inf. Sci..

[29]  Jesús Medina,et al.  Characterizing reducts in multi-adjoint concept lattices , 2018, Inf. Sci..

[30]  Jan Konecny.  Note on the characterization and reduction of concept lattices through matroid theory , 2015, Inf. Sci..

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

[32]  Luis E. Zárate,et al.  Handling high dimensionality contexts in formal concept analysis via binary decision diagrams , 2018, Inf. Sci..

[33]  Prem Kumar Singh,et al.  M-polar Fuzzy Graph Representation of Concept Lattice , 2018, Eng. Appl. Artif. Intell..

[34]  Nida Meddouri,et al.  A New Feature Selection Method for Nominal Classifier based on Formal Concept Analysis , 2017, KES.

[35]  Michal Krupka,et al.  Basic theorem of fuzzy concept lattices revisited , 2018, Fuzzy Sets Syst..

[36]  Hao Zhang,et al.  A fast incremental algorithm for deleting objects from a concept lattice , 2015, Knowl. Based Syst..

[37]  Samir Elloumi,et al.  Using minimal generators for composite isolated point extraction and conceptual binary relation coverage: Application for extracting relevant textual features , 2016, Inf. Sci..

[38]  Vilém Vychodil,et al.  Fast algorithm for computing fixpoints of Galois connections induced by object-attribute relational data , 2012, Inf. Sci..

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

[40]  Anselmo Peñas,et al.  Supporting scientific knowledge discovery with extended, generalized Formal Concept Analysis , 2016, Expert Syst. Appl..

[41]  Jan Konecny,et al.  On biconcepts in formal fuzzy concept analysis , 2017, Inf. Sci..

[42]  Radomír Halas,et al.  Generalized one-sided concept lattices with attribute preferences , 2015, Inf. Sci..