Let $(P,\leq)$ be a finite poset and let $H$ be the (acyclic) digraph of its Hasse diagram. The cut space of $H$ (also called the cocycle space of $H$) is the row space of its incidence matrix (over some field, say $\mathbb{Q}$). It is a $|V|-1$ dimensional vector space. A cut in $H$ is the edge set between a $2$-partition of the vertices of $H$, every cut has an element in $\{0,\pm 1\}^{E}$ associated with it (by choosing one side of the cut as positive). The cut space is spanned by the cuts of $H$, justifying its name. The elements in the cut space with integer values form a lattice in the cut space.
I am interested in the subset $L$ of vectors in the cut space of $H$ in which all the entries are positive integers ($L$ is infinite because the graph is acyclic). More specifically, I'm interested in those vectors in $L$ which are minimal with respect to the product order on $L$ ($u_e\leq v_e$ for every edge $e$ in $H$). This set is clearly finite (the network simplex algorithm will generate a minimal element in $L$ for example). A cut is called directed if all of its edges are in the same direction between the parts of the partition of the cut. $L$ is not a vector space but it can be spanned (nicely) by the minimal (with respect to inclusion) directed cuts of $H$.
All the reference I found on this topic is in the settings of a general digraph. I found no reference dealing with the cut space of a poset or even the cut space of an acyclic graph in general.
I was wondering if there is some reference (books or papers) in which the cut space (or its dual cycle space) of a poset or an acyclic graph are studied.