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Lattice uniformities generated by filters. (English) Zbl 0907.06015
In an earlier paper [Ann. Mat. Pura Appl. 160, 347-370 (1991; Zbl 0790.06006) and ibid. 165, 133-158 (1993; Zbl 0799.06014)], the second author studied uniform lattices as a common generalization of topological Boolean rings and topological Riesz spaces. Lattice uniformities play a similar role in the study of modular functions as FN-topologies.
The main aim of this paper is to give a contribution to the examination of the lattice structure of the space \({\mathcal L}{\mathcal U}_c(L)\) of all exhaustive lattice uniformities on a lattice \(L\). In particular, in the case where \(L\) is a modular sectionally complemented lattice, we get a satisfactory result; in this case \({\mathcal L}{\mathcal U}_c(L)\) is a complete Boolean algebra (see Corollary 5.11). Such a result can be used as the main tool in obtaining decomposition theorems for modular functions as well as information about the structure of uniform completions of \(L\).
Important for our investigations is the fact that a lattice uniformity on a sectionally complemented lattice \(L\) is uniquely determined by its \(0\)-neighbourhood system; moreover the filters which are \(0\)-neighbourhood filters for lattice uniformities are precisely the “distributive” filters, the space of which we indicate by \({\mathcal F}{\mathcal N}{\mathcal D}(L)\).
We study this space \({\mathcal F}{\mathcal N}{\mathcal D}(L)\) also for an arbitrary lattice \(L\). Every filter of \({\mathcal F}{\mathcal N}{\mathcal D}(L)\) induces on \(L\) lattice uniformity in a similar way that a distributive ideal in \(L\) induces a congruence relation. In fact, the concept of “distributive” filters introduced here and some statements about them were suggested by the concept and some properties of distributive ideals.
The main result essentialy says that, for a Hausdorff exhaustive lattice uniformity \(w\) on \(L\) generated by some \({\mathcal F}_0\in{\mathcal F}{\mathcal N}{\mathcal D}(L)\), the space of all lattice uniformities generated by filters of \({\mathcal F}{\mathcal N}{\mathcal D}(L)\) coarser than \({\mathcal F}_0\) is isomorphic to the space of all distributive elements of the completion of \((L, w)\). As a consequence, we get the result 5.11 mentioned above and the fact proved earlier [H. Weber, Order 12, 295-305 (1995; Zbl 0834.06013)] that the space of all exhaustive lattice uniformities on an orthomodular lattice is a complete Boolean algebra.

MSC:
06F30 Ordered topological structures (aspects of ordered structures)
54E15 Uniform structures and generalizations
54H12 Topological lattices, etc. (topological aspects)
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[12] H. Weber
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