发表机构
Laboratoire de Recherche LATAO Département de Mathématiques Faculté des Sciences de Tunis Université de Tunis El Manar; Preparatory Institute for Engineering Studies of Tunis University of Tunis（突尼斯埃尔马纳尔大学突尼斯科学学院数学系LATAO研究实验室; 突尼斯大学突尼斯工程预科研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解答Boulabiar关于截断Riesz空间的三个问题：截断带投影等价于主投影，单位性由不动点上确界刻画，并给出截断同态非Riesz同态的反例及其模无穷小的修正。
AI 中文摘要
我们解答了Boulabiar关于截断Riesz空间综述中的三个问题。首先，截断带投影性质与主投影性质等价，且无需完备性或Archimedean假设，此时到截断带上的带投影可由截断本身显式给出。其次，任意Riesz空间上的截断是单位性的当且仅当其不动点集有上确界。因此，$KB$-空间上的有界截断是单位性的，而带有有界截断的非单位性截断Banach格包含$c_0$的闭子格副本，且绝不是$AL$-空间；在$AM$侧，最小的截断单位化范数总是$M$-范数，但最大的未必是。第三，定义域截断为Archimedean的截断同态不一定是Riesz同态，如由$c_0$和$c_0/c_{00}$构造的显式反例所示，但当陪域空间为Archimedean时，该同态在截断无穷小意义下是Riesz同态。
英文摘要
We answer three questions from Boulabiar's survey on truncated Riesz spaces. First, the truncation band projection property coincides with the principal projection property, with no completeness or Archimedean hypothesis needed, and the band projection onto a truncation band is then given explicitly by the truncation itself. Second, a truncation on any Riesz space is unital exactly when its fixed-point set has a supremum. Hence bounded truncations on $KB$-spaces are unital, and a nonunital truncated Banach lattice with bounded truncation contains a closed sublattice copy of $c_0$ and is never an $AL$-space, while on the $AM$ side the smallest truncation unitization norm is always an $M$-norm, though the largest one need not be. Third, a truncation homomorphism whose domain truncation is Archimedean need not be a Riesz homomorphism, as an explicit counterexample built from $c_0$ and $c_0/c_{00}$ shows, but it is one modulo truncation infinitesimals whenever the codomain space is Archimedean.
