The Semantic Web of Things enhances the Internet of Things with Web technologies as well as Knowledge Graphs and reasoning. Traditional reasoners are too heavy in terms of memory footprint and/or processing time to be implementable on things. In this work, we present LiRoT, a lightweight incremental reasoner that can be embedded in constrained objects, so that reasoning on them in a fog architecture becomes possible. The focus of this work is to reduce drastically memory footprint while paying attention to processing time, hence usual optimization techniques are not fully adequate. We provide evaluations that (i) compare our system to the state of the art and (ii) show the effective benefits of the different optimizations we have implemented.