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A characterization theorem for the canonical basis of a closure operator. (English) Zbl 0959.06005

J. L. Guigues and V. Duquenne [“Familles minimales d’implications informatives résultant d’un tableau de données binaires”, Math. Sci. Hum. 95, 5-18 (1986)] defined the canonical basis of a closure operator \(\varphi\) as a special minimal generating system of \(\varphi\). They showed that all bases of \(\varphi\) can be derived from the canonical basis.
In this paper the author presents the following characterization of the canonical basis of a closure operator \(\varphi\) on a finite set \(S\):
The set \(\{(A_i, B_i)^m_{i=1}\}\) of \(m\) ordered pairs of subsets of \(S\) is equal to the canonical basis \({\mathcal B}_\varphi\) of \(\varphi\) if and only if the three following conditions are satisfied:
(1) \(\forall i\leq m\), \(A_i\subset B_i\subseteq S\),
(2) \(\forall i,j\leq m\), \((A_i\subset A_j\Rightarrow B_i\subset A_j)\),
(3) \(\forall i,j\leq m\), \((A_i\subseteq B_j \Rightarrow B_i\subseteq B_j)\).

MSC:

06A15 Galois correspondences, closure operators (in relation to ordered sets)
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