Abstract
Split-decomposition theory deals with relations between \({\mathbb R}\)-valued split systems and metrics. Here, we generalize (parts of) this theory, considering group-valued split systems that take their values in an arbitrary abelian group, and replacing metrics by certain, appropriately defined maps (some of which appear to exhibit a decidedly algebraic flavour). In the second and the third parts of this series of papers, the main results of split-decomposition theory will be established within this conceptual framework.
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J. Backelin and A.W.M. Dress, The kernel of the split map, in preparation.
Backelin J. and Linusson S. (2006). Parity splits by triple point distances in X-trees. Ann. Combin. 10: 1–18
Bandelt H.-J. (1990). Recognition of tree metrics. SIAM J. Discrete Math. 3(1): 1–6
Bandelt H.-J. and Dress A.W.M. (1992). A canonical split decomposition theory for metrics on a finite set. Adv. Math. 92(1): 47–105
Bandelt H.-J. and Dress A.W.M. (1986). Reconstructing the shape of a tree from observed dissimilarity data. Adv. Appl. Math. 7(3): 309–343
Bandelt H.-J. and Dress A.W.M. (1992). Split decomposition: a new and useful approach to phylogenetic analysis of distance data. Mol. Phylogenet Evol. 1(3): 242–252
Bandelt H.-J. and Dress A.W.M. (1989). Weak hierarchies associated with similarity measures—an additive clustering technique, Bull. Math. Biol. 51(1): 133–166
Bandelt H.-J. and Steel M.A. (1995). Symmetric matrices representable by weighted trees over a cancellative abelian monoid SIAM J. Discrete Math. 8(4): 517–525
Barker G.M. (2002). Phylogenetic diversity: a quantitative framework for measurement of priority and achievement in biodiversity conservation. Biol. J. Linnean Soc. 76(2): 165–194
Böcker S. and Dress A.W.M. (1998). Recovering symbolically dated, rooted trees from symbolic ultrametrics. Adv. Math. 138(1): 105–125
H. Colonius and H.H. Schultze, Trees constructed from empirical relations, Braunschweiger Berichte aus dem Institut fuer Psychologie 1, Braunschweig, 1977.
Devauchelle C., Dress A., Grossmann A., Grünewald S. and Henaut A. (2004). Constructing Hierarchical set systems. Ann. Combin. 8(4): 441–456
A.W.M. Dress, Split decomposition over an abelian group, part 2: group-valued split systems with weakly compatible support, Discrete Appl. Math. 157 (10) 2349–2360.
A.W.M. Dress, Split decomposition over an abelian group, part 3: group-valued split systems with compatible support, Manuscript.
Dress A.W.M. and Erdös P. (2003). X-trees and weighted quartet systems. Ann. Combin. 7(2): 155–169
Dress A.W.M. et al (2005). Δ additive and Δ ultra-additive maps, Gromov’s trees and the Farris transform. Discrete Appl. Math. 146(1): 51–73
Dress A.W.M., Huber K. and Moulton V. (2007). Some uses of the Farris transform in mathematics and phylogenetics —a review. Ann. Combin. 11(1): 1–37
Dress A.W.M. and Steel M.A. (2006). Mapping edge sets to splits in trees: the path index and parsimony. Ann. Combin. 10(1): 77–96
A.W.M. Dress and M.A. Steel, Phylogenetic diversity over an abelian group, 11 (2) (2007) 143–160.
Evans S.N. and Speed T.P. (1993). Invariants of some probability models used in phylogenetic inference. Ann. Statist. 21(1): 355–377
Faith D.P. (1992). Conservation evaluation and phylogenetic diversity. Biol. Conserv. 61(1): 1–10
J.S. Farris, On the phenetic approach to vertebrate classification, In: Major Patterns in Vertebrate Evolution, M.K. Hecht, P.C. Goody, and B.M. Hecht, Eds., Plenum Press, New York, (1977) pp. 823–850.
Farris J.S. (1979). The information content of the phylogenetic system. Sys. Zool. 28(4): 483–519
Farris J.S., Kluge A.G. and Eckardt M.J. (1970). A numerical approach to phylogenetic systematics. Sys. Zool. 19(2): 172–189
J. Felsenstein, Inferring Phylogenies, Sinauer Press, Sunderland, 2004.
Fitch W.M. and Margoliash E. (1967). Construction of phylogenetic trees. Science 155: 279–284
Heiser W.J. and Bennani M. (1997). Triadic distance models: aximomatization and least squares representation. J. Math. Psych. 41(2): 189–206
Joly S. and Calvé Le (1995). Three-way distances. J. Classification 12(2): 191–205
Korte B., Lovász L. and Schrader R. (1991). Greedoids, Algorithms and Combinatorics. Springer-Verlag, Berlin
Pachter L. and Speyer D. (2004). Reconstructing trees from subtree weights. Appl. Math. Lett. 17(6): 615–621
Semple C. and Steel M.A. (2004). Cyclic permutations and evolutionary trees. Adv. Appl. Math. 32(4): 669–680
Semple C. and Steel M.A. (2003). Phylogenetics. Oxford University Press, Oxford
Steel M.A. (2005). Phylogenetic diversity and the greedy algorithm. Syst. Biol. 54(4): 527–529
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Partly supported by the Science Technology Commission of Shanghai Municipality (Grant 06ZR14048).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Dress, A. Split Decomposition over an Abelian Group Part 1: Generalities. Ann. Comb. 13, 199–232 (2009). https://doi.org/10.1007/s00026-009-0020-2
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DOI: https://doi.org/10.1007/s00026-009-0020-2