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In this study, the authors study a two-user Gaussian doubly dirty compound multiple-access channel with partial side information (GDD-CMAC-PSI) where two independent additive interference signals are considered, each one known non-causally and partially to one of the encoders but unknown to either of the receivers. This channel, first, can model two users communicating with two base stations suffering from interference, and second, includes many previously studied channels as its special cases. For such a communication scenario, first, a general capacity outer bound is derived. Depending on the values of cross link gains, they classify the channel into three classes: weak, strong and mixed GDD-CMAC-PSI. Next, assuming that the interference signals have infinite variances, they obtain capacity outer bounds for these classes. Then, an achievable sum-rate is derived for the GDD-CMAC-PSI using Costa's strategy and thereby, they show that when both interference signals have infinite variances, this achievable sum-rate vanishes. Later, by utilising the lattice strategies and deriving achievable rate regions, independent of the interference powers, they show that in contrast with Costa's strategy, lattice-strategies can achieve positive rates. Finally, depending on signal-to-noise ratio gaps at receivers, various achievable rates are obtained.