A note on complementary knowledge spaces

Abstract

The pair $(Q,\mathcal{K})$ is a {\it knowledge space} if $\bigcup \mathcal{K}=Q$ and $\mathcal{K}$ is closed under union, where $Q$ is a nonempty set and $\mathcal{K}$ is a family of subsets of $Q$. A knowledge space $(Q,\mathcal{K})$ is called {\it complementary} if there exists a non-discrete knowledge space $(Q,\mathcal{L})$ such that the following (i) and (ii) satisfy: (i) for any $q\in Q$, there are finitely many $K_1,\cdots ,K_n\in \mathcal{K}$ and $L_1,\cdots ,L_m\in \mathcal{L}$ such that $$(\bigcap _{i=1}^n{K}_i)\cap (\bigcap _{j=1}^m{L}_j)=\{q\};$$ (ii) $\mathcal{K}\cap \mathcal{L}=\{\varnothing ,Q\}$. 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.

5 pages