Infinite Lexicographic Products

We generalize the lexicographic product of first-order structures by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times. We then define dense substructures in infinite products and show that any countable product of countable transitive homogeneous structures has a unique countable dense substructure, up to isomorphism. Furthermore, this dense substructure is transitive, homogeneous and elementarily embeds into the product. This result is then utilized to construct a rigid elementarily indivisible structure.Comment: 20 pages, 3 figure

Invariant measures concentrated on countable structures

Let L be a countable language. We say that a countable infinite L-structure M admits an invariant measure when there is a probability measure on the space of L-structures with the same underlying set as M that is invariant under permutations of that set, and that assigns measure one to the isomorphism class of M. We show that M admits an invariant measure if and only if it has trivial definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary finite tuple of M fixes no additional points. When M is a Fraisse limit in a relational language, this amounts to requiring that the age of M have strong amalgamation. Our results give rise to new instances of structures that admit invariant measures and structures that do not.Comment: 46 pages, 2 figures. Small changes following referee suggestion