Representation of ideals of relational structures

Abstract

The \textit{age} of a relational structure $\mathfrak A$ of signature $μ$ is the set $age(\mathfrak A)$ 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. If the structures are made of 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 provide an example of an ideal $I$ of isomorphism types of at most countable structures whose signature consists of a single ternary relation symbol. This ideal does not come from the set $\age_{\mathfrak I}(\mathfrak A)$ of isomorphism types of substructures of $\mathfrak A$ induced on the members of an ideal $\mathfrak I$ of sets. This answers a question due to R. Cusin and J.F. Pabion (1970).