Equimorphy: the case of chains

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 many 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.