In this paper, using the idea of ultrametrization of metric spaces we introduce ultradiversification of diversities. We show that every diversity has an ultradiversification which is the greatest nonexpansive ultra-diversity image of it. We also investigate a Hausdorff-Bayod type problem in the setting of diversities, namely, determining what diversities admit a subdominant ultradiversity. This gives a description of all diversities which can be mapped onto ultradiversities by an injective nonexpansive map. The given results generalize similar results in the setting of metric spaces.
 J. M. Bayod, J. Martínez-Maurica, Subdominant ultrametrics. Proc. Amer. Math. Soc. 109 (1990), no. 3, 829–834.Search in Google Scholar
 D. Bryant, P. F. Tupper, Hyperconvexity and tight-span theory for diversities. Adv. Math. 231 (2012), no. 6, 3172–3198.Search in Google Scholar
 D. Bryant, P. F. Tupper, Diversities and the geometry of hypergraphs. Discrete Math. Theor. Comput. Sci. 16 (2014), no. 2, 1–20.Search in Google Scholar
 D. Bryant, P. F. Tupper, Constant distortion embeddings of symmetric diversities. Anal. Geom. Metr. Spaces 4 (2016), no. 1, 326–335.Search in Google Scholar
 D. Bryant, A. Nies, P. F. Tupper, A universal separable diversity. Anal. Geom. Metr. Spaces 5 (2017), no. 1, 138–151.Search in Google Scholar
 A. J. Lemin, On ultrametrization of general metric spaces. Proc. Amer. Math. Soc. 131 (2003), no. 3, 979–989.Search in Google Scholar
 A. J. Lemin, Proximity on isosceles spaces, Russian Math. Surveys 39 (1984), no. 1, 143–144.Search in Google Scholar
 G. H. Mehrabani, K. Nourouzi, Ultradiversities and their spherical completeness. J. Appl. Anal. 26 (2020), no. 2, 231–240.Search in Google Scholar
© 2020 Pouya Haghmaram et al., published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.