Abstract
We study lattice theoretical properties of lattices of uniformities such as modularity, distributive laws and the existence of (relative) complements. For this the concepts of permutable uniformities (see Definition 3.1) and independent uniformities (see Definition 4.1) are important. Moreover, we show that e.g. the lattice of all lattice uniformities on a lattice L is a closed sublattice of the lattice of all uniformities on L.
Similar content being viewed by others
References
Avallone, A.: Lattice uniformities on orthomodular structures. Math. Slovaca 51, 403–419 (2001)
Avallone, A., Vitolo, P.: Decomposition and control theorems in effect algebras. Sci. Math. Japan 58, 1–14 (2003)
Avallone, A., Vitolo, P.: Lattice uniformities on effect algebras. Int. J. Theoret. Phys. 44, 793–806 (2005)
Avallone, A., Weber, H.: Lattice uniformities generated by filters. J. Math. Anal. Appl. 209, 507–528 (1997)
Barbieri, G., Weber, H.: A topological approach to the study of fuzzy measures. Functional analysis and economic theory (Samos, 1996), pp. 17–46. Springer, Berlin (1998)
Barbieri, G., Weber, H.: Measures on clans and on MV-algebras. Handbook of measure theory, vol. II, pp. 911–945. North-Holland, Amsterdam (2002)
Basile, A., Traynor, T.: Monotonely Cauchy locally solid topologies. Order 7, 407–416 (1991)
Basile, A., Weber, H.: Topological Boolean rings of first and second category. Seperating points for a countable family of measures, Radovi Matematički 2, 113–125 (1986)
Birkhoff, G.: Lattice theory. Amer. Math. Soc. Colloquium Publ. 30 (1967)
Bourbaki, N.: General Topology, Part 1. Hermann (1966)
Czech, E.: Topological spaces. Academia, Publishing house of the Czechoslovak Academy of Sciences. Prague (1966)
Dvurećenskij, A., Pulmannová, S.: New trends in Quantum Structures. Mathematics and its Applications, 516. Kluwer Academic Publishers (2000)
Fleischer, I., Traynor, T.: Equivalence of group-valued measures on an abstract lattice. Bull. Acad. Pol. Sci. Sér. Sci. Math. 28, 549–556 (1980)
Grätzer, G.: General lattice theory. Pure Appl. Math. 75. Academic (1978)
Graziano, M.G.: Uniformities of Fréchet-Nikodym type on Vitali spaces. Semigroup Forum 61, 91–115 (2000)
Jarchow, H.: Locally convex spaces. Teubner, Stuttgart (1981)
Klee, V.: Exotic topologies for linear spaces. General Topology and its Relations to Modern Analysis and Algebra, Proc. Symp. (Prague 1961), Prague (1962)
Kowalsky, H.J.: Beiträge zur topologischen Algebra. Math. Nachr. 11, 143–185 (1954)
Palko, V.: Topological difference posets. Math. Slovaca 47, 211–220 (1997)
Rowan, W.H.: Algebras with a compatible uniformity. Algebra Univers. 47, 13–43 (2002)
Traynor, T.: The Lebesgue decomposition for group valued set functions. Trans. Amer. Math. Soc. 220, 307–319 (1976)
Waelbroeck, L.: Topological vector spaces and algebras. Lecture Notes in Mathematics vol. 230. Springer-Verlag, Berlin (1971)
Warner, S.: Topological Fields. North-Holland Mathemathics studies 157. North-Holland, Amsterdam (1989)
Weber, H.: Zu einem Problem von Kowalsky, H.J., Abh. Braunschweig. Wiss. Gesellsch. 24, 127–134 (1978)
Weber, H.: Unabhängige Topologien, Zerlegung von Ringtopologien. Math. Z. 180, 379–393 (1982)
Weber, H.: Topological Boolean rings. Decomposition of finitely additive set functions Pac. J. Math. 110, 471–495 (1984)
Weber, H.: Uniform lattices I: a generalization of topological Riesz spaces and topological Boolean rings. Uniform lattices II: order continuity and exhaustivity. Ann. Mat. Pura Appl. 160, 347–370 (1991); 165, 133–158 (1993)
Weber, H.: Lattice uniformities and modular functions on orthomodular lattices. Order 12, 295–305 (1995)
Weber, H.: On modular functions. Funct. Approx. 24, 35–52 (1996)
Weber, H.: Uniform lattices and modular functions. Atti Sem. Mat. Fis. Univ. Modena 47, 159–182 (1999)
Weber, H.: Complemented uniform lattices. Topol. Appl. 105, 47–64 (2000)
Weber, H.: FN-topologies and group-valued measures. Handbook of measure theory, vol. I, pp. 703–743. North-Holland, Amsterdam (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Weber, H. On Lattices of Uniformities. Order 24, 249–276 (2007). https://doi.org/10.1007/s11083-007-9071-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-007-9071-3