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.
@article{118552,
author = {Josef Ml\v cek},
title = {Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {34},
year = {1993},
pages = {23-32},
zbl = {0792.03034},
mrnumber = {1240200},
language = {en},
url = {http://dml.mathdoc.fr/item/118552}
}
Mlček, Josef. Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) pp. 23-32. http://gdmltest.u-ga.fr/item/118552/
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