  When can two sequential steps performed by a computing device be considered
(causally) independent? This is a relevant question for concurrent and
distributed systems, since independence means that they could be executed in
any order, and potentially in parallel. Equivalences identifying rewriting
sequences which differ only for independent steps are at the core of the theory
of concurrency of many formalisms. We investigate the issue in the context of
the double pushout approach to rewriting in the general setting of adhesive
categories. While a consolidated theory exists for linear rules,which can
consume, preserve and generate entities, this paper focuses on left-linear
rules which may also "merge" parts of the state. This is an apparently minimal,
yet technically hard enhancement,since a standard characterisation of
independence that - in the linear case - allows one to derive a number of
properties, essential in the development of a theory of concurrency, no longer
holds. The paper performs an in-depth study of the notion of independence for
left-linear rules: it introduces a novel characterisation of independence,
identifies well-behaved classes of left-linear rewriting systems,and provides
some fundamental results including a Church-Rosser property and the existence
of canonical equivalence proofs for concurrent computations. These results
properly extends the class of formalisms that can be modelled in the adhesive
framework
