# Infinite Lexicographic Products

## TL;DR

This paper extends the concept of lexicographic products to infinite iterations, characterizes dense substructures in such products, and constructs a rigid elementarily indivisible structure with unique properties.

## Contribution

It introduces a framework for infinite lexicographic products, characterizes their dense substructures, and constructs a new rigid, elementarily indivisible structure.

## Key findings

- Any countable product of countable transitive homogeneous structures has a unique dense substructure.
- The dense substructure is transitive, homogeneous, and elementarily embeds into the product.
- Constructs a rigid elementarily indivisible structure.

## Abstract

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.