论文标题
晶格均匀性引起无界收敛
Lattice uniformities inducing unbounded convergence
论文作者
论文摘要
在本地固体riesz space $（x，τ）$中的net $（x_γ）_ {γ\inγ} $被认为是无限的$τ$ -Convergent至$ x $，如果$ |x_γ-x |x_γ-x |x_γ-x | We recall that there is a locally solid linear topology $\mathfrak{u}τ$ on $X$ such that unbounded $τ$-convergence coincides with $\mathfrak{u}τ$-convergence, and moreover, $\mathfrak{u}τ$ is characterised as the weakest locally solid linear topology which coincides with $τ$ on all order bounded subsets.正是我们引入的动机，对于均匀的晶格$（l，u）$，这是$ l $上最弱的晶格均匀性$ u^\ ast $，与$ l $的所有订单有限的子集相吻合。结果表明，如果$ u $是由本地固体riesz space $（x，τ）$的拓扑引起的统一性，则$ u^*$ - 拓扑与$ \ mathfrak {u}τ$相吻合。这允许将本文的结果与未绑定的$τ$ -Convergence上的较早结果进行比较。可以看出，尽管在统一晶格的设置中，大多数机器在[M. A. Taylor 2019：局部实心矢量空间中的无界拓扑和UO-Convergence，J。Math。肛门。应用。 \ bf {472} No.1，981--1000]缺乏“无界收敛”的概念，很好地概括了统一的晶格。我们还将回答问题2.13、3.3、5.10 [M. A. Taylor 2019：局部实心矢量空间中的无界拓扑和UO-Convergence，J。Math。肛门。应用。 \ bf {472}第1号，981--1000]和[M.的问题18.51 A. Taylor 2018：艾伯塔省论文大学矢量晶格中无限的融合]。
A net $(x_γ)_{γ\inΓ}$ in a locally solid Riesz space $(X,τ)$ is said to be unbounded $τ$-convergent to $x$ if $|x_γ-x|\wedge u\mathop{\oversetτ{\longrightarrow}} 0$ for all $u\in X_+$. We recall that there is a locally solid linear topology $\mathfrak{u}τ$ on $X$ such that unbounded $τ$-convergence coincides with $\mathfrak{u}τ$-convergence, and moreover, $\mathfrak{u}τ$ is characterised as the weakest locally solid linear topology which coincides with $τ$ on all order bounded subsets. It is with this motivation that we introduce, for a uniform lattice $(L,u)$, the weakest lattice uniformity $u^\ast$ on $L$ that coincides with $u$ on all the order bounded subsets of $L$. It is shown that if $u$ is the uniformity induced by the topology of a locally solid Riesz space $(X,τ)$, then the $u^*$-topology coincides with $\mathfrak{u}τ$. This allows comparing the results of this paper with earlier results on unbounded $τ$-convergence. It will be seen that despite the fact that in the setup of uniform lattices most of the machinery used in the techniques of [M. A. Taylor 2019: Unbounded topologies and uo-convergence in locally solid vector spaces, J. Math. Anal. Appl. \bf{472} no.1, 981--1000] is lacking, the concept of `unbounded convergence' well fittingly generalizes to uniform lattices. We shall also answer Questions 2.13, 3.3, 5.10 of [M. A. Taylor 2019: Unbounded topologies and uo-convergence in locally solid vector spaces, J. Math. Anal. Appl. \bf{472} no.1, 981--1000] and Question 18.51 of [M. A. Taylor 2018: Unbounded convergence in vector lattices, Thesis University of Alberta].
