# zbMATH — the first resource for mathematics

Control and separating points of modular functions. (English) Zbl 0965.28004
Let $$L$$ be a lattice, $$X$$ a locally convex linear space and $$\mu \:L\to X$$ a modular function, i.e., a function satisfying $\mu (x\vee y)+\mu (x\wedge y)=\mu (x)+\mu (y) .$ Let $$u(\mu)$$ denote the $$\mu$$-uniformity, i.e., the weakest uniformity on $$L$$ which makes the lattice operations and $$\mu$$ uniformly continuous. In the first part of the paper the authors study the problems, (i) when for a modular function $$\mu \:L\to X$$ there is a real valued modular function $$\nu$$ on $$L$$ with $$u(\nu)=u(\mu)$$ and (ii) when for a set $$M$$ of $$X$$-valued modular functions there is a modular function $$\nu \: L\to X$$ with $$u(\nu)$$=sup$$\{u(\mu)\:\mu \in M\}$$. Such a function $$\nu$$ is called a control for $$\mu$$ or for $$M$$, respectively. The authors present several results concerning controls on complemented lattices generalizing known results about control measures on Boolean algebras. In the second part of the paper the authors study the problem when a sequence $$\mu_n$$ of group-valued modular functions on a complemented lattice $$L$$ has a separating point, i.e., a point $$a\in L$$ such that $$\mu_n(a)\neq \mu_m(a)$$ for $$n\neq m$$. The presented results were obtained by A. Basile and H. Weber [Rad. Mat. 2, 113-125 (1986; Zbl 0596.28015)] in the case of $$L$$ being a Boolean algebra.
Reviewer: Hans Weber (Udine)

##### MSC:
 28B05 Vector-valued set functions, measures and integrals 06B30 Topological lattices 06C15 Complemented lattices, orthocomplemented lattices and posets 28B10 Group- or semigroup-valued set functions, measures and integrals
Full Text:
##### References:
  AAVALLONE A.: Liapunov theorem for modular functions. Internat. J. Theoret. Phys. 34 (1995), 1197-1204. · Zbl 0841.28007 · doi:10.1007/BF00676229  AVALLONE A.: Nonatomic vector-valued modular functions. Preprint (1995). · Zbl 0987.28012  AVALLONE A., LEPELLERE M. A.: Modular functions: Uniform boundedness and compactness. Rend. Circ. Mat. Palermo (2) XLVII (1998), 221-264. · Zbl 0931.28009 · doi:10.1007/BF02844366  AVALLONE A.-WEBER H.: Lattice uniformities generated by filters. J. Math. Anal. Appl. 209 (1997), 507-528. · Zbl 0907.06015 · doi:10.1006/jmaa.1996.5291  BARBIERI G.-WEBER H.: A topological approach to the study of fuzzy measures. Functional Analysis and Economic Theory (Y. Abramovich et al., Springer, Berlin, 1998, pp. 17-46. · Zbl 0916.28015  BASILE A.-WEBER H.: Topological Boolean rings of first and second category. Separating points for a countable family of measures, Radovi Mat. 2 (1986), 113-125. · Zbl 0596.28015  BASILE A.: Controls of families of finitely additive functions. Ricerche Mat. XXXV (1986), 291-302. · Zbl 0648.28007  BIRKHOFF G.: Lattice Theory. (3rd, Amer. Math. Soc, Providence, R.L, 1967. · Zbl 0153.02501  COSTANTINESCU C.: Some properties of speces of measures. Atti Sem. Mat. Fis. Univ. Modena 35, Supplemento (1987).  DREWNOWSKI L.: Topological rings of sets, continuous set functions, integration. III. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 20 (1972), 439-445. · Zbl 0249.28006  DREWNOWSKI L.: On control submeasures and measures. Studia Math. L (1974), 203-224. · Zbl 0285.28015 · eudml:217889  FLEISCHER I.-TRAYNOR T.: Equivalence of group-valued measure on an abstract lattice. Bull. Acad. Polon. Sci. Ser. Sci. Math. 28 (1980), 549-556. · Zbl 0514.28004  FLEISCHER I.-TRAYNOR T.: Group-valued modular functions. Algebra Universalis 14 (1982), 287-291. · Zbl 0458.06004 · doi:10.1007/BF02483932  GOULD G. G.: Extensions of vector-valued measures. Proc. London. Math. Soc. 16 (1966), 685-704. · Zbl 0148.38102 · doi:10.1112/plms/s3-16.1.685  G.: General Lattice Theory. Pure Appl. Math., Academic Press, Boston, MA, 1978. · Zbl 0436.06001  KINDLER J.: A Mazur-Orlicz type theorem for submodular set functions. J. Math. Anal. Appl. 120 (1986), 533-546. · Zbl 0605.28004 · doi:10.1016/0022-247X(86)90175-7  KRANZ P.: Mutual equivalence of vector and scalar measures on lattices. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astr. Phys. 25 (1977), 243-250. · Zbl 0361.46041  KLEMENT E. P.-WEBER S.: Generalized measure. Fuzzy Sets and Systems 40 (1991), 375-394. · Zbl 0733.28012 · doi:10.1016/0165-0114(91)90166-N  LIPECKI Z.: A characterization of group-valued measures satisfying the countable chain condition. Colloq. Math. 11 (1974), 231-234. · Zbl 0271.28009  OHBA S.: Some remarks on vector measures. Bull. Math. Soc. Sci. Math. 16 (1972), 217-223. · Zbl 0275.28015  PAP E.: Hewitt-Yosida decomposition for \square -decomposable measures. Tatra Mt. Math. Publ. 3 (1993), 147-154. · Zbl 0799.28015  PAP E.: Decompositions of supermodular functions and \square -decomposable measures. Fuzzy Sets and Systems 65 (1994), 71-83. · Zbl 0859.28012 · doi:10.1016/0165-0114(94)90248-8  SCHMIDT K.: Jordan Decompositions of Generalized Vector Measures. Pitman Res. Notes Math. Ser. 214., Longman Sci. Tech., Harlow, 1989. · Zbl 0692.28004  TRAYNOR T.: Modular functions and their Frechet-Nikodym topologies. Lecture Notes in Math. 1089, Springer, New Yourk, 1984, pp. 171-180. · Zbl 0576.28014  WEBER H.: Uniform lattices I: A generalization of topological Riesz space and topological Boolean rings; Uniform lattices II. Ann. Mat. Pura Appl. 160; 165 (1991; 1993), 347-370; 133-158. · Zbl 0790.06006 · doi:10.1007/BF01764134  WEBER H.: Valuations on complemented lattices. Internat. J. Theoret. Phys. 34 (1995), 1799-1806. · Zbl 0843.06005 · doi:10.1007/BF00676294  WEBER H.: Lattice uniformities and modular functions on orthomodular lattices. Order 12 (1995), 295-305. · Zbl 0834.06013 · doi:10.1007/BF01111744  WEBER H.: On modular functions. Funct. Approx. Comment. Math. 24 (1996), 35-52. · Zbl 0887.06011  WEBER H.: Uniform lattices and modular functions. Atti Sem. Mat. Fis. Univ. Modena XLVII (1999), 159-182. · Zbl 0989.28007  WEBER H.: Manuscript. · Zbl 1239.00005
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.