论文标题
关于脱节，频段和投影部分有序的向量空间
On disjointness, bands and projections in partially ordered vector spaces
论文作者
论文摘要
脱节，频段和频段投影是向量晶格结构理论的经典和重要组成部分。如果$ x $是这样的晶格，那些概念乍一看似乎与$ x $的格子操作密切相关。但是，过去的十五年已经看到了所有这些概念的扩展到更大的有序向量空间。 实际上，如果$ x $是具有生成锥体的阿基米德式订购的向量空间，或者是稍大的pre-riesz空间的成员，则可以给出适当的含义，并引起非平凡的结构理论。 本说明的目的是双重的：（i）我们表明，在任何riesz空间中，所有频段投影的空间的结构都非常接近我们在向量晶格的情况下所拥有的。特别是，这个空间是布尔代数。 （ii）我们给出了瑞斯前空间的几个标准，使其已经成为矢量晶格。这些标准是根据不一致性和紧密相关的概念所创造的，它们标志着在理论崩溃到矢量晶格案例之前，前雷斯空间的顺序结构如何才能获得。
Disjointness, bands, and band projections are a classical and essential part of the structure theory of vector lattices. If $X$ is such a lattice, those notions seem - at first glance - intimately related to the lattice operations on $X$. The last fifteen years, though, have seen an extension of all those concepts to a much larger class of ordered vector spaces. In fact if $X$ is an Archimedean ordered vector space with generating cone, or a member of the slightly larger class of pre-Riesz spaces, then the notions of disjointness, bands and band projections can be given proper meaning and give rise to a non-trivial structure theory. The purpose of this note is twofold: (i) We show that, on any pre-Riesz space, the structure of the space of all band projections is remarkably close to what we have in the case of vector lattices. In particular, this space is a Boolean algebra. (ii) We give several criteria for a pre-Riesz space to already be a vector lattice. These criteria are coined in terms of disjointness and closely related concepts, and they mark how lattice-like the order structure of pre-Riesz spaces can get before the theory collapses to the vector lattice case.
