Left absorption in products of countable orders

TL;DR

This paper classifies countable linear orders that are invariant under certain lexicographic products, identifying conditions for self-isomorphism and embedding properties, thus advancing understanding of order structure and self-similarity.

Contribution

It provides a complete classification of countable orders with left absorption properties and characterizes when such orders are self-similar or contain disjoint copies.

Findings

Identifies all countable orders with a non-trivial order $A$ such that $AX o X$

Determines all orders $A$ for which $AX o X$ holds

Characterizes countable orders that embed two disjoint convex copies of themselves

Abstract

We classify the countable linear orders $X$ for which there is an order $A$ with at least two points such that the lexicographic product $AX$ is isomorphic to $X$. Given such an $X$, we determine every corresponding order $A$, and identify when $X$ is isomorphic to its square. More generally, we characterize the countable orders that embed at least two disjoint convex copies of themselves.