We show that it is equivalent, for certain sets of finite graphs, to be
definable in CMS (counting monadic second-order logic, a natural extension of
monadic second-order logic), and to be recognizable in an algebraic framework
induced by the notion of modular decomposition of a finite graph. More
precisely, we consider the set F_β of composition operations on graphs
which occur in the modular decomposition of finite graphs. If F is a subset
of F_β, we say that a graph is an \calF-graph if it can be
decomposed using only operations in F. A set of F-graphs is recognizable if
it is a union of classes in a finite-index equivalence relation which is
preserved by the operations in F. We show that if F is finite and its
elements enjoy only a limited amount of commutativity -- a property which we
call weak rigidity, then recognizability is equivalent to CMS-definability.
This requirement is weak enough to be satisfied whenever all F-graphs are
posets, that is, transitive dags. In particular, our result generalizes Kuske's
recent result on series-parallel poset languages