The Median Stabilization Degree of a Median Algebra
H.-J. Bandelt
and M. van de Vel
Mathematics Seminar, Universit\ddot at Hamburg, D 2000 Hamburg 13, Bundesrepublik Deutschland
DOI: 10.1023/A:1018689708341
Abstract
The median stabilization degree (msd, for short) of a median algebra measures the largest possible number of steps needed to generate a subalgebra with an arbitrary set of generators. We determine the value of msd of a graphic n-cube Qn and we derive an estimation of msd for the natural median operator of Rn which is sharp up to one or two units. Interestingly, msd of Qn and of Rn grows like log1.5n. Finally, we characterize median algebras and median graphs of msd 
1 in terms of forbidden subspaces.
1 in terms of forbidden subspaces.Pages: 115–127
Keywords: convex structure; graphic cube; median algebra; median stabilization degree; superextension
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References
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2. H.-J. Bandelt and M. van de Vel, “Superextensions and the depth of median graphs,” J. Combinatorial Theory Series A 57 (1991), 187-202.
3. G. Birkhoff and S.A. Kiss, “A ternary operator in distributive lattices,” Bull. Amer. Math. Soc. 53 (1947), 749-752.
4. J. Eckhoff, “Der Satz von Radon in konvexen Produktstrukturen II,” Monatsh. Math. 73 (1969), 17-30.
5. E. Evans, “Median lattices and convex subalgebras,” Universal Algebra, Colloq. Math. Soc. János Bolyai 29, North-Holland, 1982, pp. 225-240.
6. C.F. Mills and W.H. Mills, “The calculation of λ(8),” 1980, preprint.
7. M. Sholander, “Trees, lattices, order, and betweenness,” Proc. Amer. Math. Soc. 3 (1952), 369-381.
8. M. Sholander, “Medians, lattices and trees,” Proc. Amer. Math. Soc. 5 (1954), 808-812.
9. M. van de Vel, “Matching binary convexities,” Top. Appl. 16 (1983), 207-235.
10. M. van de Vel, Theory of Convex Structures, North-Holland Math. Library, Vol. 50, Elsevier Science Publishers, Amsterdam, (1993), 540+xv pp.
11. A. Verbeek, Superextensions of Topological Spaces, Math. Centre Tracts, Vol. 41, Mathematisch Centrum, Amsterdam, 1972, 155 pp.
12. E.R. Verheul, Multimedians in Metric and Normed Spaces, CWI Tract, Vol. 91, Centrum voor Wiskunde en Informatika, Amsterdam, The Netherlands, 1993.