# Article

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Keywords:
monotonic valuations; ideal; semigroup
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.
References:
[M1] Mlček J.: Approximations of $\sigma$-classes and $\pi$-classes. Comment. Math. Univ. Carolinae (1979), 20 669-679. MR 0555182
[M2] Mlček J.: Valuations of structures. Comment. Math. Univ. Carolinae (1979), 20 681-695. MR 0555183
[M3] Mlček J.: Monotonic valuations and valuations of triads of higher types. Comment. Math. Univ. Carolinae (1981), 22 377-398. MR 0620373
[V] Vopěnka P.: Mathematics in the Alternative Set Theory. Teubner Texte, Leipzig, 1979. MR 0581368

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