ABSTRACT

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.

摘要

有限分布格和有限部分有序集（姿势）之间的一对一对应关系是G. Birkhoff的一个著名定理。这意味着可以通过其对应的位姿很好地表示任何分布晶格，其中前者（分布晶格）的大小通常与后者（位姿）的基础集合的大小成指数关系。许多工程和经济应用为我们带来了分布晶格作为一组环的集合，相对于集合的联合和相交而言，它们是封闭的。当涉及到一组环集时，基础集被划分为子集（或组件），并且在分区上我们有一个poset结构。这是Birkhoff定理的一个集理论变体，揭示了基础集合的分区上有限环族和有限位姿之间的对应关系，这是Masao Iri在1978年左右推行的，特别关注所谓的离散系统的主要分区，例如图，拟阵和多拟阵。在本文中，我们根据有符号环族研究了Birkhoff-Iri分解的有符号集版本，这与Reiner在有符号位姿上的结果相对应，这是Birkhoff定理的有符号对应项。我们表明，给定一个带符号的环族，我们在基础集上有一个带符号的分区，在带符号的分区上有一个带符号的波塞，代表给定的带符号的环族。这种表示在某些思考之前是唯一的。在本文中，我们根据有符号环族研究了Birkhoff-Iri分解的有符号集版本，这与Reiner在有符号位姿上的结果相对应，这是Birkhoff定理的有符号对应项。我们显示给定一个带符号的环族，我们有一个基础集的带符号的分区，并且在带符号的分区上有一个带符号的波塞，代表给定的带符号的环族。这种表示在某些思考之前是唯一的。在本文中，我们根据有符号环族研究了Birkhoff-Iri分解的有符号集版本，这与Reiner在有符号位姿上的结果相对应，这是Birkhoff定理的有符号对应项。我们表明，给定一个带符号的环族，我们在基础集上有一个带符号的分区，并且在带符号的分区上有一个带符号的波塞，代表给定的带符号的环族。这种表示在某些思考之前是唯一的。