Hostname: page-component-77c89778f8-rkxrd Total loading time: 0 Render date: 2024-07-16T09:32:37.211Z Has data issue: false hasContentIssue false
  • English
  • Français

m-Algebraic lattices in formal concept analysis

Published online by Cambridge University Press:  14 August 2019

Zhongxi Zhang Open the ORCID record for Zhongxi Zhang [Opens in a new window] ,
Qingguo Li  and
Nan Zhang
Zhongxi Zhang*
Affiliation:
School of Computer and Control Engineering, Yantai University, Yantai, Shandong, 264005, China. Email: zhangnan0851@163.com
Qingguo Li
Affiliation:
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, China. Email: liqingguoli@aliyun.com
Nan Zhang
Affiliation:
School of Computer and Control Engineering, Yantai University, Yantai, Shandong, 264005, China. Email: zhangnan0851@163.com
*
*Corresponding author. Email: zhangzhongxi89@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The notion of an m-algebraic lattice, where m stands for a cardinal number, includes numerous special cases, such as complete lattice, algebraic lattice, and prime algebraic lattice. In formal concept analysis, one fundamental result states that every concept lattice is complete, and conversely, each complete lattice is isomorphic to a concept lattice. In this paper, we introduce the notion of an m-approximable concept on each context. The m-approximable concept lattice derived from the notion is an m-algebraic lattice, and conversely, every m-algebraic lattice is isomorphic to an m-approximable concept lattice of some context. Morphisms on m-algebraic lattices and those on contexts are provided, called m-continuous functions and m-approximable morphisms, respectively. We establish a categorical equivalence between LATm, the category of m-algebraic lattices and m-continuous functions, and CXTm, the category of contexts and mapproximable morphisms.We prove that LATm is cartesian closed whenevermis regular and m > 2. By the equivalence of LATm and CXTm, we obtain that CXTm is also cartesian closed under same circumstances. The notions of a concept, an approximable concept, and a weak approximable concept are showed to be special cases of that of an m-approximable concept.

Keywords

Formal concept analysism-algebraic latticem-approximable conceptcartesian closed categorycategorical equivalence
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Issue 10 , November 2019 , pp. 1556 - 1574
Copyright
© Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S. and Jung, A. (1994) Domain theory. In: Abramsky, S., Gabbay, D.M. and Maibaum, T.S.E. (eds.) Semantic Structures, Handbook of Logic in Computer Science, vol. 3, Oxford, Clarendon Press, 1–168.Google Scholar
Apt, K. R. and Olderog, E. R. (1983). Proof rules and transformations dealing with fairness. Science of Computer Programming 3 (1) 65–100.CrossRefGoogle Scholar
Apt, K. R. and Plotkin, G. D. (1986). Countable nondeterminism and random assignment. Journal of the Association for Computing Machinery 33 (4) 724–767.CrossRefGoogle Scholar
Dauben, J. W. (1990). Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton, New Jersey, Princeton University Press.Google Scholar
Davey, B. A. and Priestley, H. A. (2002). Introduction to Lattices and Order, Cambridge, Cambridge University Press.CrossRefGoogle Scholar
Deiser, O. (2010). On the development of the notion of a cardinal number. History and Philosophy of Logic 31 (2) 123–143.CrossRefGoogle Scholar
Ganter, B. and Wille, R. (1999). Formal Concept Analysis: Mathematical Foundations. Berlin, Springer.CrossRefGoogle Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003). Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications, vol. 93, Cambridge, Cambridge University Press.Google Scholar
Grätzer, G. (1965). On the family of certain subalgebras of a universal algebra. Indagationes Mathematicae 68 790–802.CrossRefGoogle Scholar
Grätzer, G. and Lakser, H. (1992). On congruence lattices of m-complete lattices. Journal of the Australian Mathematical Society 52 (1) 57–87.CrossRefGoogle Scholar
Grätzer, G., Johnson, P. M. and Schmidt, E. T. (1994). A representation of m-algebraic lattices. Algebra Universalis 32(1) 1–12.CrossRefGoogle Scholar
Hitzler, P., Krötzsch, M. and Zhang, G. (2006). A categorical view of algebraic lattices in formal concept analysis. Fundamenta Informaticae 74 1–29.Google Scholar
Hitzler, P. and Zhang, G. (2004). A cartesian closed category of approximable concept structures. In: Wolff, K. E., Pfeiffer, H. D. and Delugach, H. S. (eds.) Conceptual Structures at Work, vol. 3127, Berlin, Heidelberg, Springer, 170–185.CrossRefGoogle Scholar
Jia, X., Jung, A., Kou, H., Li, Q. and Zhao, H. (2015). All cartesian closed categories of quasicontinuous domains consist of domains. Theoretical Computer Science 594 143–150.CrossRefGoogle Scholar
Jung, A. (1989). Cartesian Closed Categories of Domains, CWI Tracts, vol. 66. Centrum voor Wiskunde en Informatica, Amsterdam.Google Scholar
Kang, X. and Miao, D. (2016). A study on information granularity in formal concept analysis based on concept-bases. Knowledge-Based Systems 105 147–159.CrossRefGoogle Scholar
Lee, S. O. (1988). On countably approximating lattices. Bulletin of the Korean Mathematical Society 25 (1) 11–23.Google Scholar
Liu, H., Li, Q. and Guo, L. (2011). Weak approximable concepts and completely algebraic lattices. In: Li, S. (ed.) Nonlinear Mathematics for Uncertainty and its Applications, Berlin, Heidelberg, Springer, 683–689.CrossRefGoogle Scholar
MacLane, S. (1971). Categories for the Working Mathematician, New York, Springer-Verlag.Google Scholar
Nielsen, M., Plotkin, G. and Winskel, G. (1981). Petri nets, event structures and domains, part I. Theoretical Computer Science 13 (1) 85–108.CrossRefGoogle Scholar
Park, D. (1980). On the semantics of fair parallelism. In: Bjorner, D., (eds.) Abstract Software Specifications, Lecture Notes in Computer Science, vol. 86, 504–526. Springer, Berlin Heidelberg, 504–526.CrossRefGoogle Scholar
Raney, G. N. (1952). Completely distributive complete lattices. Proceedings of the American Mathematical Society 3 (5) 677–680.CrossRefGoogle Scholar
Ren, R., Wei, L. and Yao, Y. (2017). An analysis of three types of partially-known formal concepts. International Journal of Machine Learning and Cybernetics. doi: 10.1007/s13042-017-0743-z.CrossRefGoogle Scholar
Yu, H., Li, Q. and Cai, M. (2018). Characteristics of three-way concept lattices and three-way rough concept lattices. Knowledge-Based Systems 146 181–189.CrossRefGoogle Scholar
Zhang, G. and Shen, G. (2006) Approximable concepts, chu spaces, and information systems. Theory and Applications of Categories 17 (5) 79–102.Google Scholar
Zhang, Z. and Li, Q. (2017). On cartesian closed extensions of non-pointed domains. Theoretical Computer Science 691 1–9.CrossRefGoogle Scholar
Zhao, B. and Zhou, Y. (2006). The category of supercontinuous posets. Journal of Mathematical Analysis and Applications 320 (2) 632–641.CrossRefGoogle Scholar