DNA data and quasi-median algebras

Any three finite sequences (over some alphabet) weighted by, say, 3/7, 2/7, and 2/7 possess a unique sequence z minimizing their weighted Hamming distance to z. This ternary "majority" operation subject to a few equations determines what is called a quasi-median algebra, which in the finite case can be displayed as a quasi-median network. These algebras, first studied in 1980 by Martyn Mulder in a graph-theoretic context and by John Isbell from a geometric point of view, fit into an algebraic scheme for which Heinrich Werner and Brian A. Davey developed natural dualities in a series of papers in 1982-1986. Plosčica then in 1992 concretely established a full duality between quasi-median algebras and a relational ("strong compatibility") structure of partitions of a set, which in a way anticipated some concepts and results of Dress et al. and Bandelt et al. concerning the combinatorial analysis of DNA data. This duality between networks and tables of aligned sequences governs a number of dual pairs of structural features and counting formulae, which can be used, implemented in computer programs, e.g. in the applied context of quality assessment of DNA sequences (- joint work with people from the Institut für Gerichtliche Medizin, Institut für Mathematik, Innsbruck).