On elementary theories of semilattices of partial orders on sets (Q1272003)

In Algebra Univers. 28, No. 3, 324-338 (1991; Zbl 0743.03026), \textit{C. Naturman} and \textit{H. Rose} asked whether the model \(\langle A; \text{Ord} A; P^{3}\rangle\), with two basic sets, is interpretable in the semilattice \(O(A)=\langle \text{Ord} A; \cap\rangle\). Here \(\text{Ord} A\) is the collection of all partial orders on a set \(A\) and, for \(a,b,c, \in A\cup\text{Ord} A\), the predicate \(P^{3}\) is realized by the triple \(\langle a,b,c\rangle\) if and only if \(a,b\in A\), \(c\in\text{Ord} A\), and \(\langle a,b\rangle\in c\). The author gives a positive answer to the question. As a consequence, it is proven that, for arbitrary sets \(A\) and \(B\), the semilattices \(O(A)\) and \(O(B)\) are elementarily equivalent if and only if the theories of \(A\) and \(B\) (in the empty language) coincide in the complete second-order logic. A similar result is also proven for semilattices of quasiorders.