Previous |  Up |  Next

Article

Keywords:
multilattices; modular functions
Summary:
We prove that every modular function on a multilattice $L$ with values in a topological Abelian group generates a uniformity on $L$ which makes the multilattice operations uniformly continuous with respect to the exponential uniformity on the power set of $L$.
References:
[1] A. Avallone: Liapunov theorem for modular functions. Internat. J.  Theoret. Phys. 34 (1995), 1197–1204. DOI 10.1007/BF00676229 | MR 1353662 | Zbl 0841.28007
[2] A. Avallone: Nonatomic vector-valued modular functions. Annal. Soc. Math. Polon. Series I: Comment. Math. XXXIX (1999), 37–50. MR 1739014 | Zbl 0987.28012
[3] A.  Avallone, G. Barbieri and R. Cilia: Control and separating points of modular functions. Math. Slovaca 43 (1999), . MR 1696950
[4] A.  Avallone and A.  De  Simone: Extensions of modular functions on orthomodular lattices. Italian J. Pure Appl. Math, To appear. MR 1842373
[5] A.  Avallone and M. A. Lepellere: Modular functions: Uniform boundedness and compactness. Rend. Circ. Mat. Palermo XLVII (1998), 221–264. MR 1633479
[6] A.  Avallone and J. Hamhalter: Extension theorems (vector measures on quantum logics). Czechoslovak Math. J.  46 (121) (1996), 179–192. MR 1371699
[7] A.  Avallone and H. Weber: Lattice uniformities generated by filters. J.  Math. Anal. Appl. 209 (1997), 507–528. DOI 10.1006/jmaa.1996.5291 | MR 1474622
[8] G. Barbieri and H.  Weber: A topological approach to the study of fuzzy measures. Funct. Anal. Econom. Theory, Springer, 1998, pp. 17–46. MR 1730117
[9] G. Barbieri, M. A. Lepellere and H. Weber: The Hahn decomposition theorem and applications. Fuzzy Sets Systems 118 (2001), 519–528. MR 1809398
[10] H. J. Bandelt, M.  Van  de  Vel and E.  Verheul: Modular interval spaces. Math. Nachr. 163 (1993), 177–201. DOI 10.1002/mana.19931630117 | MR 1235066
[11] M. Benado: Les ensembles partiellement ordonnes et le theoreme de raffinement de Schrelier II. Czechoslovak Math.  J. 5 (1955), 308–344. MR 0076744
[12] G. Birkhoff: Lattice Theory, Third edition. AMS Providence, R.I., 1967. MR 0227053
[13] I. Fleischer and T.  Traynor: Equivalence of group-valued measure on an abstract lattice. Bull. Acad. Pol. Sci. 28 (1980), 549–556. MR 0628641
[14] I. Fleischer and T.  Traynor: Group-valued modular functions. Algebra Universalis 14 (1982), 287–291. DOI 10.1007/BF02483932 | MR 0654397
[15] M. G. Graziano: Uniformities of Fréchet-Nikodým type on Vitali spaces. Semigroup Forum 61 (2000), 91–115. DOI 10.1007/PL00006017 | MR 1839217 | Zbl 0966.28008
[16] D. J. Hensen: An axiomatic characterization of multilattices. Discrete Math. 33 (1981), 99–101. DOI 10.1016/0012-365X(81)90263-6 | MR 0597233
[17] J. Jakubík: Sur les axiomes des multistructures. Czechoslovak Math.  J. 6 (1956), 426–430. MR 0099933
[18] J. Jakubík and M.  Kolibiar: Isometries of multilattice groups. Czechoslovak Math.  J. 33 (1983), 602–612. MR 0721089
[19] J. Lihová: Valuations and distance function on directed multilattices. Math. Slovaca 46 (1996), 143–155. MR 1426999
[20] J.  Lihová and K. Repasky: Congruence relations on and varieties of directed multilattices. Math. Slovaca 38 (1988), 105–122. MR 0945364
[21] T. Traynor: Modular functions and their Fréchet-Nikodým topologies. Lectures Notes in  Math. 1089 (1984), 171–180. DOI 10.1007/BFb0072613 | MR 0786696 | Zbl 0576.28014
[22] H. Weber: Uniform Lattices I: A generalization of topological Riesz space and topological Boolean rings; Uniform lattices II. Ann. Mat. Pura Appl. 160 (1991), 347–370. MR 1163215
[23] H. Weber: Lattice uniformities and modular functions on orthomodular lattices. Order 12 (1995), 295–305. DOI 10.1007/BF01111744 | MR 1361614 | Zbl 0834.06013
[24] H. Weber: On modular functions. Funct. Approx. XXIV (1996), 35–52. MR 1453447 | Zbl 0887.06011
[25] H. Weber: Valuations on complemented lattices. Internat. J.  Theoret. Phys. 34 (1995), 1799–1806. DOI 10.1007/BF00676294 | MR 1353726 | Zbl 0843.06005
[26] H. Weber: Complemented uniform lattices. Topology Appl. 105 (2000), 47–64. DOI 10.1016/S0166-8641(99)00049-8 | MR 1761086 | Zbl 1121.54312
[27] H. Weber: Two extension theorems. Modular functions on complemented lattices. Preprint. MR 1885457 | Zbl 0998.06006
Partner of
EuDML logo