形式背景的基与基数及其性质

张琴; 魏玲

doi:10.16152/j.cnki.xdxbzr.2026-01-013

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形式背景的基与基数及其性质

粗糙集与形式概念分析中的约简研究 | 更新时间：2026-02-24

- 形式背景的基与基数及其性质
- Bases and cardinality of formal contexts and their associated properties
- 西北大学学报（自然科学版） 2026年56卷第1期页码：141-150
- 作者机构：
  
  1.西北大学　数学学院，陕西　西安　710127
  2.西北大学　概念、认知与智能研究中心，陕西　西安　710127
- 作者简介：
  
  张琴，女，博士研究生，从事形式概念分析等研究，xzzonly1@163.com。
  魏玲，女，教授，博士生导师，从事形式概念分析、粒计算、三支决策理论等研究，wl@nwu.edu.cn。
- 基金信息：
  
  国家自然科学基金(12171392);陕西数理基础科学研究项目(23JSZ008)
- DOI：10.16152/j.cnki.xdxbzr.2026-01-013
  中图分类号： O29;TP18
- 收稿：2025-07-01，
  
  修回：2025-08-07，
  
  纸质出版：2026-02-25
- 稿件说明：
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张琴, 魏玲. 形式背景的基与基数及其性质[J]. 西北大学学报（自然科学版）, 2026,56(1):141-150.

ZHANG Qin, WEI Ling. Bases and cardinality of formal contexts and their associated properties[J]. Journal of Northwest University (Natural Science Edition), 2026, 56(1): 141-150.
张琴, 魏玲. 形式背景的基与基数及其性质[J]. 西北大学学报（自然科学版）, 2026,56(1):141-150. DOI： 10.16152/j.cnki.xdxbzr.2026-01-013.

ZHANG Qin, WEI Ling. Bases and cardinality of formal contexts and their associated properties[J]. Journal of Northwest University (Natural Science Edition), 2026, 56(1): 141-150. DOI： 10.16152/j.cnki.xdxbzr.2026-01-013.

摘要

保持二元关系不变的概念约简是形式概念分析中的一个重要研究方向，可在不丢失原始信息的前提下仅选择一部分概念进行知识发现。一个形式背景的不同概念约简中的概念个数可能不同，其中概念个数最少的概念约简表明用最少的概念就能恢复原始背景的所有二元关系，可极大地简化知识表示方式。首先，给出形式背景的基与基数、最膨胀概念约简及最压缩概念约简的定义，并研究它们的性质；其次，研究形式背景的基与对象概念集（属性概念集）的关系，继而给出对象概念集（属性概念集）是基的等价刻画；最后，从形式背景的Ferrers维数与集合维数角度给出形式背景的基数范围。

Abstract

Concept reduction preserving binary relation is an important research direction in formal concept analysis

which allows for knowledge discovery by selecting some formal concepts (not all) without losing original information. The number of concepts in different concept reducts of a formal context may vary

where the concept reducts with the fewest number of formal concepts indicate that all binary relations can be recovered by the fewest formal concepts. This greatly simplifies the knowledge representation. We first define the bases and the cardinality of a formal context

the most expanded concept reducts and the most compressed concept reducts

and investigate their properties. Then

we explore the relationship between the bases of formal context and the set of object concepts (the set of attribute concepts)

and provide an equivalent condition for when the set of object concepts (the set of attribute concepts) serves as a basis. Finally

we present the range of the cardinality of a formal context from the perspectives of the set dimension and the Ferrers dimension of a formal context.

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references

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