zbMATH — the first resource for mathematics

Valuations on complemented lattices. (English) Zbl 0843.06005
It is proved that the space of all bounded real-valued valuations \(m\) with \(m(0)= 0\) on a complemented lattice is isomorphic to the space of all real-valued totally additive measures on a suitable complete Boolean algebra. This answers a question of P. Pták affirmatively. The proof is based on a Hahn-decomposition theorem and an extension theorem for valuations.

06C15 Complemented lattices, orthocomplemented lattices and posets
28A12 Contents, measures, outer measures, capacities
06E10 Chain conditions, complete algebras
Full Text: DOI
[1] Avallone, A., and Weber, H. (n.d.). Lattice uniformities generated by filters, preprint. · Zbl 0907.06015
[2] Birkhoff, G. (1984).Lattice Theory, American Mathematical Society, Providence, Rhode Island. · Zbl 0063.00402
[3] Fleischer, I., and Traynor, T. (1982). Group-valued modular functions,Algebra Universalis,14, 287-291. · Zbl 0487.06002 · doi:10.1007/BF02483932
[4] Gr?tzer, G. (1978).General Lattice Theory, Academic Press, New York. · Zbl 0436.06001
[5] Kiseleva, T. G. (1967). Partially ordered sets endowed with a uniform structure,Vestnik Leningrad University,22(13), 51-57 [in Russian].
[6] Maeda, F., and Maeda, S. (1970).Theory of Symmetric Lattices, Springer-Verlag, Berlin. · Zbl 0219.06002
[7] Sikorski, R. (1969).Boolean Algebras, Springer-Verlag, Berlin. · Zbl 0191.31505
[8] Weber, H. (1991). Uniform lattices I: A generalization of topological Riesz spaces and topological Boolean rings,Annali di Matematica Pura e Applicata,160, 347-370. · Zbl 0790.06006 · doi:10.1007/BF01764134
[9] Weber, H. (1993). Uniform lattices II: Order continuity and exhaustivity,Annali di Matematica Pura e Applicata,165, 133-158. · Zbl 0799.06014 · doi:10.1007/BF01765846
[10] Weber, H. (n.d.-a). Lattice uniformities and modular functions on orthomodular lattices,Order (to appear). · Zbl 0834.06013
[11] Weber, H. (n.d.-b). On modular functions,Funct. et Approx. (to appear). · Zbl 0887.06011
[12] Weber, H. (n.d.-c). Independent lattice uniformities, manuscript.
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.