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Keywords:
poset; semilattice; tree; semimodularity; chain condition; height; distance; metric; triangle inequality
poset; semilattice; tree; semimodularity; chain condition; height; distance; metric; triangle inequality
Summary:
Discrete partially ordered sets can be turned into distance spaces in several ways. The distance functions may or may not satisfy the triangle inequality and restrictions of the distance to finite chains may or may not coincide with the natural, difference-of-height distance measured in a chain. It is shown that for semilattices the semimodularity ensures the good behaviour of the distances considered. The Jordan-Dedekind chain condition, which is weaker than semimodularity, is equivalent to the basic criterion that the graph-theoretic distance (realized by zig-zagging up and down freely in the poset to connect two points) is compatible with distances measured on chains by the relative height. Semimodularity is shown to be equivalent to the validity of the triangle inequality of a restricted graph-theoretic distance, called the up-down distance. The fact that the up-down distance corresponds to the computation of degrees of kinship in family trees leads to the observation that the less familiar canon-law method of computation corresponds also to a mathematically well behaved Chebyshev-type distance on discrete semilattices. For the Chebyshev distance also semimodularity is shown to imply the validity of the triangle inequality. The reverse implication fails, but assuming the validity of the triangle inequality, the semimodularity is shown to have a local characterization by a forbidden six-element subsemilattice. Like in the classical case of real spaces, the Chebyshev semilattice distance is shown to be the limit of a converging sequence of distances, all of them verifying the triangle inequality if the semilattice is semimodular.
References:
[1] Bouchard, C. B.: Consanguinity and noble marriages in the tenth and eleventh centuries. Speculum 56 (1981), 268-287. DOI 10.2307/2846935
[2] Burtsell, R.: Canonical adoption. The Catholic Encyclopedia Vol. 1 Robert Appleton Company, New York (1907). Available at http://www.newadvent.org/cathen/01147b.htm
[3] Chartrand, G., Johns, G. L., Tian, S., Winters, S. J.: Directed distance in digraphs: Centers and medians. J. Graph Theory 17 (1993), 509-521. DOI 10.1002/jgt.3190170408 | MR 1231014 | Zbl 0781.05018
[4] Deza, M. M., Panteleeva, E.: Quasi-semi-metrics, oriented multi-cuts and related polyhedra. Eur. J. Comb. 21 (2000), 777-795. DOI 10.1006/eujc.1999.0383 | MR 1791206 | Zbl 0966.52010
[5] Foldes, S., Woodroofe, R.: Antichain cutsets of strongly connected posets. Order 30 (2013), 351-361. DOI 10.1007/s11083-012-9248-2 | MR 3063192 | Zbl 1282.06009
[6] Garner, B. A.: A Dictionary of Modern Legal Usage. Oxford University Press, Oxford (2001).
[7] Haskins, L., Gudder, S.: Height on posets and graphs. Discrete Math. 2 (1972), 357-382. DOI 10.1016/0012-365X(72)90014-3 | MR 0306059 | Zbl 0238.06002
[8] Kharat, V. S., Waphare, B. N., Thakare, N. K.: On forbidden configurations for strong posets. Algebra Univers. 51 (2004), 111-124. DOI 10.1007/s00012-004-1851-7 | MR 2067152 | Zbl 1079.06006
[9] Monjardet, B.: Metrics on partially ordered sets--A survey. Discrete Math. 35 (1981), 173-184. DOI 10.1016/0012-365X(81)90206-5 | MR 0620670 | Zbl 0463.46016
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