## 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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