Signed ring families and signed posets
Signed ring families and signed posets
复制标题
带符号环族和带符号偏序集
DOI：
10.1080/10556788.2020.1740219
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发表时间：
2020
影响因子：
2.2
通讯作者：
Kazutoshi Ando an Satoru Fujishige
中科院分区：
文献类型：
--
作者：
Fujishige Satoru;Takazawa Kenjiro;Yokoi Yu;Kazutoshi Ando an Satoru Fujishige
有限分配格与有限偏序集之间的一一对应是G。伯克霍夫这意味着任何分配格都可以用其对应的偏序集来表示，其中前者（分配格）的大小通常与后者的基础集（偏序集）的大小成指数关系。许多工程和经济应用给我们带来了分配格作为一个环族的集合，它是封闭的关于集合的并和交。当涉及到集合的环族时，底层集合被划分为子集（或分量），并且我们在划分上有一个偏序集结构。这是伯克霍夫定理的一个集合论变体，揭示了有限环族和有限偏序集在基础集合的划分上的对应关系，这是由Masao Iri在1978年左右追求的，特别关注所谓的离散系统的主划分，如图，拟阵和多拟阵。在本文中，我们研究的Birkhoff-Iri分解的符号集版本的带符号环族，这对应于Reiner的结果，带符号偏序集，一个带符号的对应的Birkhoff定理。我们证明了，给定一个有符号的环族，我们有一个有符号的分区的基础集连同一个有符号的偏序集的签署分区，代表给定的有符号的环族。对于某些反射来说，这种表示是独特的。
The one-to-one correspondence between finite distributive lattices and finite partially ordered sets (posets) is a well-known theorem of G. Birkhoff. This implies a nice representation of any distributive lattice by its corresponding poset, where the size of the former (distributive lattice) is often exponential in the size of the underlying set of the latter (poset). A lot of engineering and economic applications bring us distributive lattices as a ring family of sets which is closed with respect to the set union and intersection. When it comes to a ring family of sets, the underlying set is partitioned into subsets (or components) and we have a poset structure on the partition. This is a set-theoretical variant of the Birkhoff theorem revealing the correspondence between finite ring families and finite posets on partitions of the underlying sets, which was pursued by Masao Iri around 1978, especially concerned with what is called the principal partition of discrete systems such as graphs, matroids, and polymatroids. In the present paper we investigate a signed-set version of the Birkhoff-Iri decomposition in terms of signed ring family, which corresponds to Reiner's result on signed posets, a signed counterpart of the Birkhoff theorem. We show that given a signed ring family, we have a signed partition of the underlying set together with a signed poset on the signed partition which represents the given signed ring family. This representation is unique up to certain reflections.