## On critical sets of a finite Moore family.(English)Zbl 1306.62139

Summary: Cluster collections obtained within the framework of most cluster structures studied in data analysis and classification are essentially Moore families. In this paper, we propose a simple intuitive necessary and sufficient condition for some subset of objects to be a critical set of a finite Moore family. This condition is based on a new characterization of quasi-closed sets. Moreover, we provide a necessary condition for a subset containing more than $$k$$ objects ($$k\geq 2$$) to be a critical set of a $$k$$-weakly hierarchical Moore family. Finally, as a consequence of this result, we identify critical sets of some $$k$$-weakly hierarchical Moore families and thereby generalize a result earlier obtained by Domenach and Leclerc in the particular case of weak hierarchies.

### MSC:

 62H30 Classification and discrimination; cluster analysis (statistical aspects) 06A15 Galois correspondences, closure operators (in relation to ordered sets) 62-07 Data analysis (statistics) (MSC2010)
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