Conserved intervals were recently introduced as a measure of similarity between genomes whose genes have been shuffled during evolution by genomic rearrangements. Phylogenetic reconstruction based on such similarity measures raises many biological, formal and algorithmic questions, in particular the labelling of internal nodes with putative ancestral gene orders, and the selection of a good tree topology. In this paper, we investigate the properties of sets of permutations associated to conserved intervals as a representation of putative ancestral gene orders for a given tree topology. We define set-theoretic operations on sets of conserved intervals, together with the associated algorithms, and we apply these techniques, in a manner similar to the Fitch-Hartigan algorithm for parsimony, to a subset of chloroplast genes of 13 species.
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