A'Campo–Gusein-Zade diagrams as partially ordered sets

The real analogues of many results about complex monodromies of singularities can be formulated and proved in terms of partial orderings on A'Campo–Gusein-Zade diagrams, the real versions of Coxeter–Dynkin diagrams of singularities. In this paper it is proved that the only diagrams among the A'Campo–Gusein-Zade diagrams of singularities that determine partially ordered sets of finite type (in the sense of representations of a quiver) are the diagrams of simple singularities. To encode the real decompositions of a singularity the analogue of Vasilev invariants turn out to be surjections of a partially ordered set onto a chain. Formulae are proved for Arnold $(\operatorname{mod}2)$-invariants of plane curves in terms of the corresponding A'Campo–Gusein-Zade diagrams. We define, in the context of higher Bruhat orders, higher partially ordered sets and we describe their connection with the higher $M$-Morsifications $A_n$. We also consider certain previously known results about real singularities from the point of view of partially ordered sets.