  The Lov\'{a}sz Local Lemma (LLL) is a powerful tool in probabilistic
combinatorics which can be used to establish the existence of objects that
satisfy certain properties. The breakthrough paper of Moser and Tardos and
follow-up works revealed that the LLL has intimate connections with a class of
stochastic local search algorithms for finding such desirable objects. In
particular, it can be seen as a sufficient condition for this type of
algorithms to converge fast.
  Besides conditions for existence of and fast convergence to desirable
objects, one may naturally ask further questions regarding properties of these
algorithms. For instance, "are they parallelizable?", "how many solutions can
they output?", "what is the expected "weight" of a solution?", etc. These
questions and more have been answered for a class of LLL-inspired algorithms
called commutative. In this paper we introduce a new, very natural and more
general notion of commutativity (essentially matrix commutativity) which allows
us to show a number of new refined properties of LLL-inspired local search
algorithms with significantly simpler proofs.
