  The Bost-Connes Hecke C^*-algebra can be regarded as a direct limit of
subalgebras involving finite sets of primes. Each of these finite-prime
analogues of the Bost-Connes algebra is a crossed product by a semigroup N^F,
where F is finite. We describe composition series of ideals for these semigroup
crossed products in which the intermediate subquotients are finite direct sums
of classifiable AT-algebras. We then use similar techniques to identify
subquotients of the Bost-Connes algebra as classifiable AT-algebras.
