有序向量空间中的序收敛：紧性、原子、普遍性与冰淇淋空间
Order convergence in ordered vector spaces: compactness, atoms, pervasiveness, and ice cream spaces
AI总结：
该研究探讨非格有序向量空间的序收敛，推广向量格的序收敛概念，证明普遍性JB-代数的结合性，刻画普遍性空间，并研究冰淇淋空间的序性质，为有序向量空间理论提供反例来源。
摘要：
我们研究不一定是格的有序向量空间中的序收敛。在此背景下，我们考虑向量格上序收敛的两种推广形式：定向变体与非定向变体。我们证明，对于普遍性空间类，这两种概念是一致的，并将向量格中关于原子性和序收敛的若干结果推广到该类空间。特别地，我们证明普遍性JB-代数是结合的，这强化了Sherman针对C*-代数的一个结果。我们还建立了普遍性空间的若干新刻画。此外，我们系统研究了与巴拿赫空间相关的一类序单位空间，称为冰淇淋空间。我们给出这类空间中定向序收敛与非定向序收敛的描述，得到两种概念存在差异的例子。我们还考察了冰淇淋空间的其他序论方面，如原子和格运算，并将其与基础巴拿赫空间的几何关联起来。我们证明冰淇淋空间表现出与向量格截然不同的行为，可作为有序向量空间理论中丰富的反例来源。
英文摘要：
We study order convergence in ordered vector spaces that are not necessarily lattices. In this setting, we consider two generalizations of order convergence on vector lattices, a directed and an undirected variant. We show that these notions agree for the class of pervasive spaces and extend several results concerning atomicity and order convergence in vector lattices to this class. In particular, we prove that a pervasive JB-algebra is associative, which strengthens a result of Sherman for C*-algebras. We also establish a number of new characterizations of pervasive spaces. Additionally, we systematically study a class of order unit spaces associated to Banach spaces, called ice cream spaces. We provide descriptions of both directed and undirected order convergences in these spaces, yielding examples where the two notions differ. We also consider other order-theoretic aspects of ice cream spaces, such as atoms and lattice operations, relating them to the geometry of the underlying Banach spaces. We demonstrate that ice cream spaces exhibit behavior very different from vector lattices and can serve as a rich source of counterexamples in the theory of ordered vector spaces.
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