Biclosed sets, quasitrivial semigroups and oriented matroid
Abstract
In this paper, we establish a one-to-one correspondence between the set of biclosed sets in an irreducible root system of type An 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 An, 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 Bn quasitrivial semigroups, and prove that they are in bijective with biclosed sets in a type Bn 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.