Geodesic
Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 1, pp. 23-32

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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.
Classification : 03E70, 20M14, 93A15
Keywords: monotonic valuations; ideal; semigroup 
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     title = {Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals},
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Mlček, Josef. Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 1, pp. 23-32. http://geodesic.mathdoc.fr/item/CMUC_1993__34_1_a2/