We consider the problem of building satisfiability procedures for unions of disjoint theories. We briefly review the combination schemas proposed by Nelson-Oppen, Shostak, and others. Three inference systems are directly derived from the properties satisfied by the theories being combined and known results from the literature are obtained in a uniform and abstract way. This rational reconstruction is the starting point for further investigations. We introduce the concept of extended canonizer and derive a modularity result for a new class of theories (larger than Shostak and smaller than Nelson-Oppen theories) which is closed under disjoint union. This is in contrast with the lack of modularity of Shostak theories. We also explain how to implement extended canonizers by using the basic building blocks used in Shostak schema or by means of rewriting techniques.
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