Differing from the conventional communication system paradigm that models information source as a sequence of (i.i.d. or stationary) random variables, the semantic approach aims at extracting and sending the high-level features of the content deeply contained in the source, thereby breaking the performance limits from the statistical information theory. As a pioneering work in this area, the deep learning-enabled semantic communication (DeepSC) constitutes a novel algorithmic framework based on the transformer--which is a deep learning tool widely used to process text numerically. The main goal of this work is to extend the DeepSC approach from the point-to-point link to the multi-user multiple access channel (MAC). The inter-user interference has long been identified as the bottleneck of the MAC. In the classic information theory, the successive interference cancellation (SIC) scheme is a common way to mitigate interference and achieve the channel capacity. Our main contribution is to incorporate the SIC scheme into the DeepSC. As opposed to the traditional SIC that removes interference in the digital symbol domain, the proposed semantic SIC works in the domain of the semantic word embedding vectors. Furthermore, to enhance the training efficiency, we propose a pretraining scheme and a partial retraining scheme that quickly adjust the neural network parameters when new users are added to the MAC. We also modify the existing loss function to facilitate training. Finally, we present numerical experiments to demonstrate the advantage of the proposed semantic approach as compared to the existing benchmark methods.