AI 中文总结
该研究引入Dedekind完备向量格的上确界-下确界完备化，发展格星框架完成其构造，确立其基本性质，将Riemann积分推广到I、II型反常积分，证明全完备向量格上幂级数可逐项积分。
AI 中文摘要
我们引入了Dedekind完备向量格的上确界-下确界完备化，这是一种本质上唯一的扩张，其中每个非空子集都有上确界和下确界。由于该完备化不是锥，我们发展了更一般的格星框架，为其构造提供自然的环境。我们确立了上确界-下确界完备化的基本性质，包括泛性质、表示定理及其带和带投影的刻画。作为应用，我们将Dedekind完备$f$-代数上的Riemann积分推广到I型和II型反常积分。最后，我们证明了全完备向量格上的幂级数可逐项积分。
英文摘要
We introduce the sup-inf-completion of a Dedekind complete vector lattice, an essentially unique extension in which every nonempty subset has both a supremum and an infimum. Since this completion is not a cone, we develop the more general framework of lattice stars, which provides the natural setting for its construction. We establish the fundamental properties of the sup-inf-completion, including a universal property, a representation theorem, and a characterization of its bands and band projections. As an application, we extend the Riemann integral on Dedekind complete $f$-algebras to Type I and Type II improper integrals. We conclude by showing that power series on universally complete vector lattices may be integrated term-by-term.
