  Motivated by q-shuffle products determined by Singer from q-analogues of
multiple zeta values, we build in this article a generalisation of the shuffle
and stuffle products in terms of weak shuffle and stuffle products. Then, we
characterise weak shuffle products and give as examples the case of an alphabet
of cardinality two or three. We focus on a comparison between algebraic
structures respected in the classical case and in the weak case. As in the
classical case, each weak shuffle product can be equipped with a dendriform
structure. However, they have another behaviour towards the quadri-algebra and
the Hopf algebra structure. We give some relations satisfied by weak stuffle
products.
