Monotonic valuations of $\pi \sigma$-triads and evaluations of ideals

Mlček, Josef. "Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals." Commentationes Mathematicae Universitatis Carolinae 34.1 (1993): 23-32. <http://eudml.org/doc/247479>.

@article{Mlček1993,
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\}\{^\{\prime \prime \}\}\lbrace r\in Q;\,r\ge 0 \& r\doteq 0\rbrace $. 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.},
author = {Mlček, Josef},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {monotonic valuations; ideal; semigroup; alternative set theory; monotonic valuations; -class; generalizations of the metrization theorem; ideals; triad},
language = {eng},
number = {1},
pages = {23-32},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals},
url = {http://eudml.org/doc/247479},
volume = {34},
year = {1993},
}

TY - JOUR
AU - Mlček, Josef
TI - Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1993
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 34
IS - 1
SP - 23
EP - 32
AB - 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}{^{\prime \prime }}\lbrace r\in Q;\,r\ge 0 \& r\doteq 0\rbrace $. 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.
LA - eng
KW - monotonic valuations; ideal; semigroup; alternative set theory; monotonic valuations; -class; generalizations of the metrization theorem; ideals; triad
UR - http://eudml.org/doc/247479
ER -