  The Markov commutator associated to a finite Markov kernel P is the convex
semigroup consisting of all Markov kernels commuting with P. Its interest comes
from its relation with the hypergroup property and with the notion of Markovian
duality by intertwining. In particular, it is shown that the discrete analogue
of the Achour-Trim{\`e}che's theorem, asserting the preservation of
non-negativity by the wave equations associated to certain Metropolis birth and
death transition kernels, cannot be extended to all convex potentials. But it
remains true for symmetric and monotone convex potentials. Keywords: finite
Markov kernels, Markov commutator, symmetry group of a Markov kernel,
hypergroup property, duality by intertwining, Achour-Trim{\`e}che theorem,
birth and death chains, Metropolis algorithms, one-dimensional discrete wave
equations.
