  We examine classes of quantum algebras emerging from involutive,
non-degenerate set-theoretic solutions of the Yang-Baxter equation and their
q-analogues. After providing some universal results on quasi-bialgebras and
admissible Drinfeld twists we show that the quantum algebras produced from
set-theoretic solutions and their q-analogues are in fact quasi-triangular
quasi-bialgebras. Specific illustrative examples compatible with our generic
findings are worked out. In the q-deformed case of set-theoretic solutions we
also construct admissible Drinfeld twists similar to the set-theoretic ones,
subject to certain extra constraints dictated by the q-deformation. These
findings greatly generalise recent relevant results on set theoretic solutions
and their q-deformed analogues.
