Lower semicontinuous type regularity conditions for subdifferential calculus

We give a lower semicontinuous type regularity condition and a closedness type one which turn out to be necessary and sufficient for the fulfilment of two different formulae involving the -subdifferential of a perturbation function, respectively. These regularity conditions prove to be sufficient also for having formulae for the classical subdifferential of a perturbation function. Some recently published results concerning -subdifferentials are rediscovered as special cases.