zbMATH — the first resource for mathematics

Uniform lattices. II: Order continuity and exhaustivity. (English) Zbl 0799.06014
This paper is a continuation of Part I [ibid. 160, 347-370 (1991; Zbl 0790.06006)]. For the definition of lattice topologies and lattice uniformities we refer to Section 1.1 of Part I.
The main aim of this paper is to give conditions under which a lattice topology is weaker than another one. For that order continuity plays a decisive role. Our general setting leads to common generalizations of results for \(FN\)-topologies on Boolean rings and locally solid topologies on Riesz spaces or on \(l\)-groups. In §5 the problem of comparing lattice topologies is examined under an order-completeness assumption on the lattice. That assumption can be dropped if one of the lattice topologies is induced by an exhaustive lattice uniformity. In Section 8.1 the problem above is examined under additional countability assumptions. In Section 8.2 we briefly point out that also Lebesgue’s dominated convergence theorem can be formulated and proved in our abstract setting as consequence of previous considerations. Some results of this paper can also be used as key in the study of modular functions, which will be explained in another paper.

06B30 Topological lattices
06F30 Ordered topological structures (aspects of ordered structures)
54H12 Topological lattices, etc. (topological aspects)
Full Text: DOI
[1] Aliprantis, C. D.; Burkinshaw, O., Locally Solid Riesz Spaces (1978), New York: Academic Press, New York · Zbl 0402.46005
[2] G.Birkhoff,Lattice Theory, AMS Colloquium Publications, vol.25, Providence, Rhode Island (1984). · JFM 66.0100.04
[3] Basile, A.; Traynor, T., Monotonely Cauchy locally-solid topologies, Order, 7, 407-416 (1991) · Zbl 0778.46007
[4] Drewnowski, L., Topological rings of sets, continuous set functions, integration I, II, III, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astr. Phys., 20, 269-276 (1972) · Zbl 0249.28004
[5] Kiseleva, T. G., Partially ordered sets endowed with a uniform structure (Russian), Vestnik Leningrad Univ., 22, No. 13, 51-57 (1967)
[6] Lawson, J. D., Intrinsic topologies in topological lattices and semilattices, Pacific J. Math., 44, 593-602 (1973) · Zbl 0253.06013
[7] Lipecki, Z., Extensions of additive set functions with values in a topological group, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astr. Phys., 22, 19-27 (1974) · Zbl 0275.28013
[8] Maharam, D., From finite to countable additivity, Portugaliae Math., 44, 265-282 (1987) · Zbl 0636.28002
[9] Strauss, D. P., Topological lattices, Proc. London Math. Soc., 18, 217-230 (1968) · Zbl 0153.33404
[10] Weber, H., Ein Fortsetzungssatz für gruppenwertige Maβe, Arch. Math., 34, 157-159 (1980) · Zbl 0422.28006
[11] Weber, H., Vergleich monotoner Ringtopologien und absolute Stetigkeit von Inhalten, Commentarii Mathematici Universitatis Sancti Pauli, 31, 49-60 (1982) · Zbl 0485.28004
[12] H.Weber,Group-and vector-valued s-bounded contents, inMeasure Theory (Oberwolfach 1983), Lecture Notes in Mathematics, vol.1089, Springer-Verlag (1984), pp. 181-198.
[13] Weber, H., Uniform lattices I. A generalization of topological Riesz space and topological Boolean rings, Ann. Mat. Pura Appl., 160, 347-370 (1991) · Zbl 0790.06006
[14] H.Weber,On minimal topologies, Adv. Math., to appear. · Zbl 0861.12005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.