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.

中文翻译：

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无限词典产品

我们通过提出一个结构框架来概括一阶结构的字典乘积，从某种意义上说，它模仿了字典乘积的无限迭代，不一定是多次迭代。然后我们在无穷积中定义稠密子结构，并证明可数传递同质结构的任何可数积都具有唯一的可数稠密子结构，直至同构。此外，这种密集的子结构是可传递的、同质的，并且基本嵌入到产品中。然后利用这个结果来构建一个刚性的基本不可分割的结构。