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.

Let L be a finite relational language. The age of a structure M over L is the set of isomorphism types of finite substructures of M. We classify those ages U for which there are less than 2ω countably infinite pairwise nonisomorphic L-structures of age U.

A poset is well‐partially ordered (WPO) if all its linear extensions are well orders; the supremum of ordered types of these linear extensions is the length, of p. We prove that if the vertex set X is infinite, of cardinality κ, and the ordering ⩽ is the intersection of finitely many well partial orderings of X,, then, letting, with, denote the euclidian division by κ (seen as an initial ordinal) of the length of each corresponding poset: where denotes the least (...) initial ordinal greater than the ordinal. This inequality is optimal. This result answers questions of Forster. (shrink)

The age of a relational structure R is the set A of finite restrictions of R considered up to isomorphism. R. Fraı̈ssé, who introduced this notion, showed that ages coincide with nonempty ideals of the poset consisting of finite relational structures, considered up to isomorphism and ordered by embeddability. Here, given two ages A ⊆ B , we study the poset D consisting of ages C in sandwich between A and B . Among other things we show that if D (...) is infinite then it contains an infinite chain. (shrink)