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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.

06B30 Topological lattices
54H12 Topological lattices, etc. (topological aspects)
06F30 Ordered topological structures (aspects of ordered structures)
Full Text: DOI
[1] Aliprantis, C. D.; Burkinshaw, O., Locally Solid Riezs Spaces (1978), New York: Academic Press, New York
[2] G.Birkhoff,Lattice Theory, AMS Colloquium Publications, vol. 25, Providence, Rhode Island (1984). · JFM 66.0100.04
[3] Basile, A.; Weber, H., Topological Boolean rings of first and second category. Separating points for a countable family of measures, Radovi Matematički., 2, 113-125 (1986) · Zbl 0596.28015
[4] Constantinescu, C., Some properties of spaces of measures, Suppl. Atti Sem. Mat. Fis. Univ. Modena, 35, 1-286 (1989)
[5] Drewnowski, L., Topological rings of sets, continuous set functions, integration I, Bull. Acad, Polon. Sci., Sér. Sci. Math. Astr. Phys., 20, 269-276 (1972) · Zbl 0249.28004
[6] Fremlin, D. H., Topological Riesz Spaces and Mesure Theory (1974), Cambridge: Cambridge University Press, Cambridge · Zbl 0273.46035
[7] Fleischer, I.; Traynor, T., Group-valued modular functions, Algebra Universalis, 14, 287-291 (1982) · Zbl 0458.06004
[8] Kiseleva, T. G., Partially ordered sets endowed with a uniform structure (Russian), Vestnik Leningrad. Univ., 22, Nr. 13, 51-57 (1967)
[9] Nachbin, L., Topology and Order (1965), New York: D. Van Nostrand Company, New York · Zbl 0131.37903
[10] Schäpke, F. W., Integrationstheorie und quasinormierte Gruppen, J. reine angew. Math., 253, 117-137 (1972) · Zbl 0233.28009
[11] Schmidt, K. D., Jordan decompositions of generalized vector measures, Pitman Research Notes in Mathematics, series, vol.214 (1989), Essex: Longman, Essex · Zbl 0692.28004
[12] Šmarda, B., The lattice of topologies of topological l-groups, Czechoslovak Math. J., 26, 128-136 (1976) · Zbl 0333.54004
[13] Weber, H., R-freie Integrationstheorie I, II, J. reine angew. Math., 289, 30-54 (1977) · Zbl 0341.28007
[14] Weber, H., Vergleich monotoner Ringtopologien und absolute Stetigkeit von Inhalten, Commentarii Mathematici Universitatis Sancti Pauli, 31, 49-60 (1982) · Zbl 0485.28004
[15] H.Weber,Metrization of uniform lattices, Czechoslovak Math. J. · Zbl 0818.06006
[16] Wilhelm, M., Completeness of l-groups and of l-seminorms, Comment. Math., 21, 271-281 (1979) · Zbl 0441.06016
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