The partial ordering on the automorphism group of the countable generic partial order (Q1047182)

The countable generic partial order \((P,\leq)\) is defined by using the well-known Fraïssé's Theorem. This paper deals with the structure \((G,\circ,\leq)\), where \((G,\circ)=\text{Aut}(P,\leq)\) and \(\leq\) is the pointwise ordering on \(G\). It is shown that \((G,\leq)\) is elementarily equivalent to \((P,\leq)\) itself and, more generally, that \((G,\circ,\leq)\) satisfies a weakening of the existential closure property for partially ordered groups. This leads to the study of the group \(G^*\) obtained by freely adjoining a finite set of generators to \(G\).