The induced path function, monotonicity and betweenness

The induced path function $J(u, v)$ of a graph consists of the set of all vertices lying on the induced paths between vertices $u$ and $v$. This function is a special instance of a transit function. The function $J$ satisfies betweenness if $w \in J(u, v)$ implies $u \notin J(w, v)$ and $x \in J(u, v)$ implies $J(u, x \subseteq J(u, v)$, and it is monotone if $x, y \in J(u, v)$ implies $J(x, y) \subseteq J(u, v)$. The induced path function of a connected graph satisfying the betweenness and monotone axioms are characterized by transit axioms.