Josef Ml\v cek
Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals

Comment.Math.Univ.Carolinae 34,1 (1993) 23-32.

Abstract:{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.}

Keywords: monotonic valuations, ideal, semigroup
AMS Subject Classification: O3E70, 93A15, 20M14

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