Amalgamation Classes with $\exists$-Resolutions

Abstract

Let $K_d$ denote the class of all finite graphs and, for graphs $A\subseteq B$, say $A \leq_d B$ if distances in $A$ are preserved in $B$; i.e. for $a, a' \in A$ the length of the shortest path in $A$ from $a$ to $a'$ is the same as the length of the shortest path in $B$ from $a$ to $a'$. In this situation $(K_d, \leq_d)$ forms an amalgamation class and one can perform a Hrushovski construction to obtain a generic of the class. One particular feature of the class $(K_d, \leq_d)$ is that a closed superset of a finite set need not include all minimal pairs obtained iteratively over that set but only enough such pairs to resolve distances; we will say that such classes have $\exists$-resolutions. Larry Moss conjectured the existence of graph $M$ which was $(K_d, \leq_d)$-injective (for $A \leq_d B$ any isometric embedding of $A$ into $M$ extends to an isometric embedding of $B$ into $M$) but without finite closures. We examine Moss's conjecture in the more general context of amalgamation classes. In particular, we will show that the question is in some sense more interesting in classes with $\exists$-resolutions and will give some conditions under which the possibility of such structures is limited.