Flag in an incidence system

Symbol-free definition
A flag in an incidence system is a set of pairwise incident elements.

Definition with symbols
Let $$(G,*,t,I)$$ be an incidence system. Then, a flag in $$G$$ is a set $$\Phi$$ of pairwise incident elements of $$G$$, viz a set $$\Phi$$ such that $$x * y$$ for any $$x,y \in \Phi$$.

Graph-theoretic definition
A flag in an incidence system is what corresponds to a clique in the corresponding partite graph.

Rank and type of a flag
The type of a flag is the set of types of all its elements. The rank of a flag is the cardinality of the flag (and also of the type, since all elements of the flag are drawn from different types).

Corank and cotype of the flag
The cotype of a flag is the complement (in the set of all types) of the type of the flag. The corank is the cardinality of the cotype.