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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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