# Signed incidence structures

I've been trying to understand GraphQL queries (e.g. Wikidata) with Formal Concept Analysis but my first stumbling block is that most between-object relations at least in the Wikidata ontology are signed, they have a "from" and a "to".

Let me maybe rewind: an incidence structure is $$(G,M,I)$$ where $$G$$ is a set of "points" and $$M$$ is a set of "lines" and $$I\subset G\times M$$ is the incidence relation. So for example $$(G,M)$$ could be a bipartite graph; or $$G$$ could be a set of edges that are incident to nodes in a simple graph. Formal concept analysis is a theory of lattices over incidence structures where $$G$$ are "concepts" and $$M$$ are "attributes".

Google finally tells me that there are higher-order incident structures with more "types of entities" (i.e. $$G_1, \cdots, G_r$$ but as best as I can tell as a nongeometer, these still have a single incidence relation. And what I need is something like $$(G,M,I\times \sigma)$$ where $$\sigma=\{-1,1\}$$ is the sign or orientation of the incidence relation.

My question is "does this exist as something with an already-existing literature"? I do know that directed graphs in network flow problems can be represented as signed incidence matrices, but I'm trying to see if I can minimally amend the ideas of formal concept analysis to accomodate signed/directed relations.

(Edit: reviewing my notes, the actual problem is somewhat worse than what I just described.

A typical signed-relation is something like "Markov taught Kolmogorov". But it's not enough to say that point-Markov is incident to line "teacher" with sign -1 (from or source) while point-Kolmogorov is incident to the same line with sign 1 (to or target). That merely establishes that Kolmogorov had teachers and Markov had students, but doesn't link both. So a signed relation is more like

$$G\times G \times \sigma, \quad M, \quad I\subset (G\times G\times M)$$

i.e. points are signed in a given signed-relation, not the incidence relation that binds them. But maybe this departs enough from incidence geometry that it's worth studying the $$(G, M, I\times \sigma)$$ "signed incidence structure" if it does come up in the literature.)