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The notion of representation systems describes structures that are algebraically closed under queries. It has recently been realized that representation systems are highly relevant also in the context of data exchange. The authors extend the notion of representation system to encompass data exchange map-pings and their composition. Seen through this lens, two major classes of representation systems emerge, namely homomorphic data exchange systems and strong data exchange systems. The homomorphic "OWA" systems encompass the "Classical" part of data exchange. Reasoning is modulo homomorphic equivalence (CQ-equivalence), and only unions of conjunctive queries and monotone data exchange map-pings are supported.
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