Geometric constructions of higher Bruhat orders and $M$-Morsifications

Higher Bruhat orders were introduced by Manin and Schechtman in the course of studying multi-dimensional generalizations of the Yang–Baxter equations. In this paper we present a problem from real singularity theory which generalizes Arnol'ds snake calculus (a coding of the connected components of the space of very nice $M$-Morsifications of a singularity $A_n$) and in which the role of updown permutations is played by their higher analogues – elements of special form in higher Bruhat orders.