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Biclosed sets, quasitrivial semigroups and oriented matroid

Weijia Wang, Rui Wang

TL;DR

The paper builds a precise bridge between root-system combinatorics and algebraic semigroup theory by establishing a Weyl-group-equivariant bijection between biclosed sets in the $A_n$ root system and quasitrivial semigroup structures on a set of size $n{+}1$. It then extends this correspondence to standard parabolic subsets, yielding a concrete enumeration for parabolic biclosed sets in type $A$, and defines an index that matches the associativity index of the corresponding quasitrivial magma. Extending the framework to type $B_n$, the authors characterize biclosed sets via type $B_n$ quasitrivial semigroups and discuss the natural symmetry action. Finally, they provide a purely combinatorial oriented matroid construction for type $A_n$ using total preorders, bypassing geometric realizations. Collectively, these results connect root-system structure with semigroup theory and offer new combinatorial tools for studying parabolic orders 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.

Biclosed sets, quasitrivial semigroups and oriented matroid

TL;DR

The paper builds a precise bridge between root-system combinatorics and algebraic semigroup theory by establishing a Weyl-group-equivariant bijection between biclosed sets in the root system and quasitrivial semigroup structures on a set of size . It then extends this correspondence to standard parabolic subsets, yielding a concrete enumeration for parabolic biclosed sets in type , and defines an index that matches the associativity index of the corresponding quasitrivial magma. Extending the framework to type , the authors characterize biclosed sets via type quasitrivial semigroups and discuss the natural symmetry action. Finally, they provide a purely combinatorial oriented matroid construction for type using total preorders, bypassing geometric realizations. Collectively, these results connect root-system structure with semigroup theory and offer new combinatorial tools for studying parabolic orders 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 and the set of quasitrivial semigroup structures on a set with 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 . Secondly, we define an index for an arbitrary subset of the root system of type , 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 quasitrivial semigroups, and prove that they are in bijective with biclosed sets in a type root system. Finally, by identifying certain biclosed sets with total preorders, we present a purely combinatorial proof that a root system of type possesses an oriented matroid structure.
Paper Structure (10 sections, 18 theorems, 119 equations, 1 figure, 4 tables)

This paper contains 10 sections, 18 theorems, 119 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Quasitrivial semigroup
  4. The group $S_n^B$
  5. Root systems of type $A_n, B_n$ and its biclosed sets
  6. The bijection
  7. bijection for biclosed sets in standard parabolic subsets
  8. Associativity index of quasitrivial magmas
  9. quasitrivial semigroup of type $B_n$
  10. Oriented matroid structure from total preorder

Key Result

Lemma 3.1

There exists a bijection between $P_{(k_1,k_2,\cdots, k_t)}$ and $Q_{t,p}$.

Figures (1)

  • Figure 1: Colored area representation of the sets involved.

Theorems & Definitions (39)

  • Lemma 3.1
  • proof
  • Example 3.2
  • Theorem 3.3
  • proof
  • Example 3.4
  • Proposition 3.5
  • proof
  • Lemma 3.6
  • proof
  • ...and 29 more