论文标题
双座套件，准半群和定向曲霉
Biclosed sets, quasitrivial semigroups and oriented matroid
论文作者
论文摘要
在本文中，我们在类型$ a_n $的不可约根系统中建立了一对一的对应关系，而在带有$ n+1 $元素的集合中，我们的类型$ a_n $的根系和一组准二依型半群结构。在此对应关系的基础上，我们首先将这种两次试验概括为在标准的抛物线子集中提供双聚体集的半群结构表征。特别是，这使我们能够以抛物线型$ a $的抛物线弱点的方式得出元素的枚举结果。其次，我们为类型$ a_n $的根系的任意子集定义了一个索引，该索引量化了其远离双裂的偏差，并证明了这种索引与相关的准杂岩的关联索引一致。第三，我们定义类型$ b_n $ quasitrivial Semogroups，并证明它们是在类型$ b_n $ root Systems中带有双斑的套件。最后，通过识别某些具有总预订的双面套件，我们提供了一个纯粹的组合证明，即$ a $类型的根系具有定向的矩形结构。
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.
