  Aims. We examine the interactions of various instabilities in rotating stars,
which usually are considered as independent. Methods. An analytical study of
the problem is performed, account is given to radiative losses, mu-gradients
and horizontal turbulence. Results. The diffusion coefficient for an ensemble
of instabilities is not given by the sum of the specific coefficients for each
instability, but by the solution of a general equation. We find that
thermohaline mixing is possible in low-mass red giants only if the horizontal
turbulence is very weak. In rotating stars the Rayleigh-Taylor and the shear
instabilities need simultaneous treating. This has for consequence that
rotation laws of the form 1/r^(alpha) are predicted to be unstable for alpha >
1.6568, while the usual Rayleigh criterion predicts instability only for alpha
> 2. Also, the shear instabilities are somehow reduced in Main Sequence stars
by the effect of the Rayleigh-Taylor criterion. Various instability criteria
should be expressed differently in rotating stars than in simplified
geometries.
