双闭集、拟平凡半群与有向拟阵
Biclosed sets, quasitrivial semigroups and oriented matroid
AI总结：
本文建立$A_n$型根系双闭集与拟平凡半群的一一对应，推广至抛物子集并给出计数，定义偏差指标，引入$B_n$型拟平凡半群，并利用双闭集与全预序的等同给出$A$型根系有向拟阵结构的纯组合证明。
摘要：
本文建立了$A_n$型不可约根系中双闭集集合与具有$n+1$个元素的集合上的拟平凡半群结构集合之间的一一对应关系。基于这一对应关系，我们首先推广该双射，以提供标准抛物子集中双闭集的半群结构刻画。特别地，这使我们能够推导出$A$型抛物弱序中元素的计数结果。其次，我们为$A_n$型根系的任意子集定义一个指标，该指标量化了它们与双闭集的偏差，并证明该指标与相关拟平凡岩浆的结合性指标一致。第三，我们定义$B_n$型拟平凡半群，并证明它们与$B_n$型根系中的双闭集双射。最后，通过将某些双闭集与全预序等同，我们给出了一个纯组合证明，表明$A$型根系具有有向拟阵结构。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
