  Weighted labelled transition systems are LTSs whose transitions are given
weights drawn from a commutative monoid. WLTSs subsume a wide range of LTSs,
providing a general notion of strong (weighted) bisimulation. In this paper we
extend this framework towards other behavioural equivalences, by considering
semirings of weights. Taking advantage of this extra structure, we introduce a
general notion of weak weighted bisimulation. We show that weak weighted
bisimulation coincides with the usual weak bisimulations in the cases of
non-deterministic and fully-probabilistic systems; moreover, it naturally
provides a definition of weak bisimulation also for kinds of LTSs where this
notion is currently missing (such as, stochastic systems). Finally, we provide
a categorical account of the coalgebraic construction of weak weighted
bisimulation; this construction points out how to port our approach to other
equivalences based on different notion of observability.
