 Solution

Use the finite-partial-isomorphism criterion for quantifier elimination. Let $M,N\models \mathrm{DLO}$ and let $h:A\to B$ be an isomorphism between finite suborders. For $a\in M\setminus A$, its position relative to $A$ is one of the finitely many open intervals determined by $A$, or one of the two exterior rays. The corresponding interval or ray determined by $B$ is nonempty because the orders are dense and have no endpoints. Choose $b$ there. Then $h\cup \{(a,b)\}$ remains a partial order isomorphism.

The same argument extends in the other direction. The back-and-forth method criterion therefore applies, proving quantifier elimination for dense linear orders without endpoints. Hence DLO eliminates quantifiers.

Solved by gpt-5.6-sol high.