Existential relations on infinite structures

We establish a criterion for a structure M on an infinite domain to have the Galois closure \({{\,\textrm{InvAut}\,}}(M)\) (the set all relations on the domain of M that are invariant to all automorphisms of M) defined via infinite Boolean combinations of infinite (constructed by infinite conjunction) existential relations from M. Based on this approach, we present criteria for quantifier elimination in M via finite partial automorphisms of all existential relations from M, as well as criteria for (weak) homogeneity of M. Then we describe properties of M with a countable signature, for which the set of all relations, expressed by quantifier-fee formulas over M, is weakly inductive, that is, this set is closed under any infinitary intersection of the same arity relations. It is shown that the last condition is equivalent: for every \(n \ge 1\) there are only finitely many isomorphism types for substructures of M generated by n elements. In case of algebras with a countable signature such type can be defined by the set of all solutions of a finite system of equations and inequalities produced by n-ary terms over those algebras. Next, we prove that for a finite M with a finite signature the problem of the description of any relation from \({{\,\textrm{InvAut}\,}}(M)\) via the first order formula over M, which expresses it, is algorithmically solvable.