Equimorphy -- The Case of Chains

TL;DR

This paper investigates the properties of chains (linear orders) that are equimorphic, establishing that each chain either has a unique or infinitely many isomorphism classes of equimorphic chains, with detailed structure results for those with fewer than continuum classes.

Contribution

It provides new structural insights into chains with fewer than continuum classes of equimorphic chains and proves a dichotomy for the number of such classes for any chain.

Findings

Chains with fewer than continuum classes have specific structural properties.

Any chain has either one or infinitely many isomorphism classes of equimorphic chains.

The paper characterizes chains based on the number of equimorphic classes they possess.

Abstract

Two structures are said to be equimorphic if each embeds in the other. Such structures cannot be expected to be isomorphic, and in this paper we investigate the special case of linear orders, here also called chains. In particular we provide structure results for chains having less than continuum any isomorphism classes of equimorphic chains. We deduce as a corollary that any chain has either a single isomorphism class of equimorphic chains or infinitely many.