  We present a unified framework for modelling genomes and their rearrangements
in a genome algebra, as elements that simultaneously incorporate all physical
symmetries. Building on previous work utilising the group algebra of the
symmetric group, we explicitly construct the genome algebra for the case of
unsigned circular genomes with dihedral symmetry and show that the maximum
likelihood estimate (MLE) of genome rearrangement distance can be validly and
more efficiently performed in this setting. We then construct the genome
algebra for a more general case, that is, for genomes that may be represented
by elements of an arbitrary group and symmetry group, and show that the MLE
computations can be performed entirely within this framework. There is no
prescribed model in this framework; that is, it allows any choice of
rearrangements that preserve the set of regions, along with arbitrary weights.
Further, since the likelihood function is built from path probabilities -- a
generalisation of path counts -- the framework may be utilised for any distance
measure that is based on path probabilities.
