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**Applications des isomorphismes HTS et HTE. (Vers la classification arboree). (Applications of the HTS and HTE isomorphisms. (Towards a free classification)).**
*(French)*
Zbl 0724.05056

Summary: In the first article [see the preceding review] we have presented the isomorphisms HTS and HET for all the dissimilarities with the usual order. The images are the couples (K,t) where K is a hypergraph and t an index on K. The movement back towards the dissimilarities is made by DIS.

When the Ks are hierarchies, HTS and HTE coincide with the BenzĂ©cri/Johnson bijection. When the Ks are pyramids, HTS (respectively HTE) is the Fichet bijection (respectively Diday) and concerns pyra dissimilarities. The pyras occurs in the pyramidal classification: the representation of a dissimilarity is shown by a dendrogram of plane visualisation.

In this second article, after the pyras come naturally a class of dissimilarities which we call the arbas because their hypergraph image is arboreal (K is arboreal if there is a tree (in French: “arbre”) for which every element of K is connected). We show that every pyra is an arba and that the arboreal classification (a dissimilarity is represented by an arba, whether it is greater, lesser or proximal) can be found in a contex of spatial visualisation. Thanks to the isomorphisms HTS, HTE and DIS, we demonstrate that for a couple (dissimilarity, tree) there is a subdominant arba and an overdominant arba.

When the Ks are hierarchies, HTS and HTE coincide with the BenzĂ©cri/Johnson bijection. When the Ks are pyramids, HTS (respectively HTE) is the Fichet bijection (respectively Diday) and concerns pyra dissimilarities. The pyras occurs in the pyramidal classification: the representation of a dissimilarity is shown by a dendrogram of plane visualisation.

In this second article, after the pyras come naturally a class of dissimilarities which we call the arbas because their hypergraph image is arboreal (K is arboreal if there is a tree (in French: “arbre”) for which every element of K is connected). We show that every pyra is an arba and that the arboreal classification (a dissimilarity is represented by an arba, whether it is greater, lesser or proximal) can be found in a contex of spatial visualisation. Thanks to the isomorphisms HTS, HTE and DIS, we demonstrate that for a couple (dissimilarity, tree) there is a subdominant arba and an overdominant arba.

### MSC:

05C65 | Hypergraphs |

62-07 | Data analysis (statistics) (MSC2010) |

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |