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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
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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 

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