The coherency index Original Research Article

A coherent decomposition of a metric is, in general, a natural decomposition of the metric into a sum of simpler metrics. A metric that has only a trivial coherent decomposition is called prime. In this paper we give a formula for an index, called the coherency index, which allows us to prove that there exist only finitely many prime metrics (up to multiplication by a positive scalar) on a finite set. Moreover, the formula that we give for the coherency index also provides us with a computational tool by which one can compute coherent decompositions of metrics into primes.