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Title: Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals (English)
Author: Mlček, Josef
Language: English
Journal: Commentationes Mathematicae Universitatis Carolinae
ISSN: 0010-2628 (print)
ISSN: 1213-7243 (online)
Volume: 34
Issue: 1
Year: 1993
Pages: 23-32
Category: math
Summary: We develop problems of monotonic valuations of triads. A theorem on monotonic valuations of triads of the type $\pi \sigma $ is presented. We study, using the notion of the monotonic valuation, representations of ideals by monotone and subadditive mappings. We prove, for example, that there exists, for each ideal $J$ of the type $\pi $ on a set $A$, a monotone and subadditive set-mapping $h$ on $P(A)$ with values in non-negative rational numbers such that $J = h^{-1}{''}\{r\in Q;\,r\geq 0 \& r\doteq 0\}$. Some analogical results are proved for ideals of the types $\sigma ,\,\sigma \pi $ and $\pi \sigma $, too. A problem of an additive representation is also discussed. (English)
Keyword: monotonic valuations
Keyword: ideal
Keyword: semigroup
MSC: 03E70
MSC: 20M14
MSC: 93A15
idZBL: Zbl 0792.03034
idMR: MR1240200
Date available: 2009-01-08T18:00:47Z
Last updated: 2012-04-30
Stable URL:
Reference: [M1] Mlček J.: Approximations of $\sigma $-classes and $\pi $-classes.Comment. Math. Univ. Carolinae (1979), 20 669-679. MR 0555182
Reference: [M2] Mlček J.: Valuations of structures.Comment. Math. Univ. Carolinae (1979), 20 681-695. MR 0555183
Reference: [M3] Mlček J.: Monotonic valuations and valuations of triads of higher types.Comment. Math. Univ. Carolinae (1981), 22 377-398. MR 0620373
Reference: [V] Vopěnka P.: Mathematics in the Alternative Set Theory.Teubner Texte, Leipzig, 1979. MR 0581368


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