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Uniform lattices. I: A generalization of topological Riesz spaces and topological Boolean rings. (English) Zbl 0790.06006
A uniform lattice is a pair $$(L,u)$$ consisting of a lattice $$L$$ and a uniformity $$u$$ on $$L$$ making the lattice operations uniformly continuous. The obvious examples are locally solid topological $$l$$-groups and locally convex topological Boolean rings, with the uniformities induced by the topologies. Several other examples (and counterexamples) are given in the paper.
The author first develops a basic theory of uniform lattices, paying special attention to convergence of Cauchy nets. Next, $$(L,u)$$ is said to satisfy the condition $$(\sigma)$$ if for every $$U\in u$$ there is a sequence $$(U_ n)$$ in $$u$$ with the following property: if $$(a_ n)$$ is a monotone sequence in $$L$$, order convergent to $$a$$ and if $$(a_ i,a_ j)\in U_ n$$ as soon as $$i,j\geq n$$, then $$(a_ 1,a)\in U$$. This condition links order convergence and $$u$$-convergence of sequences. For instance, if $$u$$ has a countable base, then $$(\sigma)$$ is equivalent to the property that for monotone Cauchy sequences order convergence coincides with $$u$$-convergence.
The author also introduces a separation property $$(L)$$ that generalizes the Fatou property for seminorms on Riesz spaces. With the aid of this property he proves a completeness theorem for uniform lattices that extends Nakano’s completeness theorem for Riesz spaces.

##### MSC:
 06B30 Topological lattices 54H12 Topological lattices, etc. (topological aspects) 06F30 Ordered topological structures (aspects of ordered structures)
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