Representation of ideals of relational structures

The age of a relational structure A of signature μ is the set a g e ( A ) of its finite induced substructures, considered up to isomorphism. This is an ideal in the poset Ω μ consisting of finite structures of signature μ and ordered by embeddability. We shall show that if the structures have infinitely many relations and if, among those, infinitely many are at least binary then there are ideals which do not come from an age. We provide many examples. We particularly look at metric spaces and offer several problems. We also answer a question due to Cusin and Pabion [R. Cusin, J.F. Pabion, Une généralisation de l’âge des relations, C. R. Acad. Sci. Paris, Sér. A-B 270 (1970) A17–A20]: there is an ideal I of isomorphism types of at most countable structures whose signature consists of a single ternary relation symbol such that I does not come from the set age I ( A ) of isomorphism types of substructures of A induced on the members of an ideal I of sets.