Biclosed sets, quasitrivial semigroups and oriented matroid

TL;DR

This paper establishes a deep connection between biclosed sets in root systems and quasitrivial semigroup structures, providing new enumeration, characterization, and combinatorial insights across types A and B.

Contribution

It introduces a bijection between biclosed sets and quasitrivial semigroups, generalizes this to parabolic subsets, and links biclosed sets with oriented matroids in type A.

Findings

Bijection between biclosed sets and quasitrivial semigroups in type A

Enumeration of elements in parabolic weak orders of type A

Identification of biclosed sets with total preorders and oriented matroids

Abstract

In this paper, we establish a one-to-one correspondence between the set of biclosed sets in an irreducible root system of type $A_n$ and the set of quasitrivial semigroup structures on a set with $n+1$ elements. Building on this correspondence, we first generalize this bijection to provide a semigroup structural characterization of the biclosed sets in a standard parabolic subset. In particular, this allows us to derive an enumeration result for the elements in a parabolic weak order of type $A$. Secondly, we define an index for an arbitrary subset of the root system of type $A_n$, which quantifies their deviation from from being biclosed, and prove that such an index coincides with the associativity index of the associated quasitrivial magma. Thirdly, we define type $B_n$ quasitrivial semigroups, and prove that they are in bijective with biclosed sets in a type $B_n$ root system. Finally, by identifying certain biclosed sets with total preorders, we present a purely combinatorial proof that a root system of type $A$ possesses an oriented matroid structure.