We show that
-rational series, and
-valued monadic second-order logic have the same expressive power, for any bounded lattice
and for finite and infinite words. This extends classical results of Kleene and Büchi to arbitrary bounded lattices, without any distributivity assumption that is fundamental in the theory of weighted automata over semirings. In fact, we obtain these results for large classes of strong bimonoids which properly contain all bounded lattices.