Dense linear order without endpoints

A dense linear order without endpoints is a linear order in which a third point lies strictly between every two distinct points and every point has points both below and above it.

Add constants $c_0<c_1<\cdots$ to the theory of dense linear orders without endpoints. The three countable models have respectively a least upper bound for the named sequence, no upper bound, or upper bounds with no least one. The unbounded model is an atomic model; the bounded model with no least upper bound is an omega-saturated model. The remaining model omits the type $c_i<x<d$ over its least-upper-bound parameter $d$.

A complete dense linear order without endpoints with the countable chain condition for a linear order, but with no countable order-dense subset.

The assertion that there is no Suslin line, equivalently that every complete dense linear order without endpoints satisfying the interval countable chain condition is separable. A Suslin tree gives a counterexample.

The theory of dense linear orders without endpoints eliminates quantifiers. A finite partial order isomorphism extends by placing each new point in the corresponding interval or ray.