On canonicity of almost linear minimal orders

Abstract

We prove that if a minimal ordered structure $(M,<, \ldots)$ with infinite chains in an arbitrary language extending the language of strict orders interprets (in some power $M^n$) the linear order $(\omega,<)$, then this fact can be witnessed for $n=1$ via the incomparability relation of an $<$-definable strict order $R$; that $R$ is unique up to ``almost equality'', i.e., finite rearrangements of elements of $M$; and that $R$ can be defined from $<$ in a constructive way. We also show that several variants of the question whether a minimal ordered structure interprets an infinite linear order are all equivalent.