# Article

 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: http://hdl.handle.net/10338.dmlcz/118552 . 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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