A note on complementary knowledge spaces

2023/08/17 by Lin, Fucai
#FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.2308.08733

Abstract

The pair (Q, \mathscrK) is a \it knowledge space if \bigcup\mathscrK=Q and \mathscrK is closed under union, where Q is a nonempty set and \mathscrK is a family of subsets of Q. A knowledge space (Q, \mathscrK) is called \it complementary if there exists a non-discrete knowledge space (Q, \mathscrL) such that the following (i) and (ii) satisfy: (i) for any q∈ Q, there are finitely many K₁, ⋯, K_n∈ \mathscrK and L₁, ⋯, L_m∈ \mathscrL such that (\bigcap_i=1ⁿK_i)∩ (\bigcap_j=1^mL_j)=\q\; (ii) \mathscrK∩ \mathscrL=\∅, Q\. In this paper, the existence of a complementary knowledge space for each knowledge space is proved, and a method of the construction of complementary finite knowledge spaces is given.

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