interesting for finite model theorists. Easy relations between the notions Clearly every basic elementary class is an elementary class, and every elementary class is a pseudo-elementary class. Moreover, as an easy consequence of the compactness theorem, a class of σ-structures is basic elementary if and only if it is elementary and its complement is also elementary. Examples A basic elementary class Let σ be a signature consisting only of a unary function symbol f. The class K of σ-structures in which f is one-to-one is a basic elementary class. This is witnessed by the theory T, which consists only of the single sentence . An elementary, basic pseudoelementary class that is not basic elementary Let σ be an arbitrary signature. The class K of all infinite σ-structures is elementary. To see this, consider the sentences "", "", and so on. (So the sentence says that there are at least n elements.) The infinite σ-structures are precisely the models of the theory . But K is not a basic elementary class. Otherwise the infinite σ-structures would be precisely those that satisfy a certain first-order sentence τ. But then the set would be inconsistent. By the compactness theorem, for some natural number n the set would be inconsistent. But this is absurd, because this theory is satisfied by any finite σ-structure with or more elements. However, there is a basic elementary class K' in the signature σ' = σ {f}, where f is a unary function symbol, such that K consists exactly of the reducts to σ of σ'-structures in K'. K' is axiomatised by the single sentence , which expresses that f is injective but not surjective. Therefore, K is elementary and what could be called basic pseudo-elementary, but not basic elementary. Pseudo-elementary class that is non-elementary Finally, consider the signature σ consisting of a single unary relation symbol P. Every σ-structure is partitioned into two subsets: Those elements for which P holds, and the rest. Let K be the class of all σ-structures for which these two subsets have the same cardinality, i.e., there is a bijection between them. This class is not elementary, because a σ-structure in which both the set of realisations of P and its complement are countably infinite satisfies precisely the same first-order sentences as a σ-structure in which one of the sets is countably infinite and the other is uncountable. Now consider the signature ,