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Let $(A,E)$ be a directed 2-uniform hypergraph and $E$ the corresponding binary relation such that $(X,Y) \in E$ iff there is a hyperedge from $X$ to $Y$. We say that there is a path from $X_1$ to $X_k$ iff there is a sequence $(X_1,...,X_k)$ such $(X_1,X_2) \in E \wedge (X_2,X_3) \in E \wedge ... \wedge (X_{k-1},X_k)\in E$.

Is there a MSO-sentence reach such $(A,E) \models \text{reach}(U,V)$ iff $V$ is reachable from $U$?

Reachability for graphs in general is definable by reach$(u,v)$ $$\exists X \forall Y ((u \in Y \wedge \forall x \forall y (x \in Y \wedge y \in X \wedge E(x,y))\rightarrow y \in Y) \rightarrow v \in Y)\text{.}$$ However, I am not successful with replacing the first order variables by second order ones (and the corresponding $\in$ by $\subseteq$).

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    $\begingroup$ Sorry, I do not understand the question. What is the difference between a "directed 2-uniform hypergraph" and simply a directed graph? What are the elements of the set $A$ in your notion of a directed 2-uniform hypergraph? Are they something else than vertices of a graph? In fact, I wonder if there isn't some underlying confusion in your question: MSO formulas can use quantification over sets, but the domain of the structure on which they are evaluated consists of elements just like in first-order logic. $\endgroup$
    – a3nm
    Jan 24 at 18:43

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