| Title:
|
Monotonic valuations of $\pi \sigma $-triads and evaluations of ideals (English) |
| Author:
|
Mlček, Josef |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
34 |
| Issue:
|
1 |
| Year:
|
1993 |
| Pages:
|
23-32 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
monotonic valuations |
| Keyword:
|
ideal |
| Keyword:
|
semigroup |
| MSC:
|
03E70 |
| MSC:
|
20M14 |
| MSC:
|
93A15 |
| idZBL:
|
Zbl 0792.03034 |
| idMR:
|
MR1240200 |
| . |
| Date available:
|
2009-01-08T18:00:47Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118552 |
| . |
| Reference:
|
[M1] Mlček J.: Approximations of $\sigma $-classes and $\pi $-classes.Comment. Math. Univ. Carolinae (1979), 20 669-679. MR 0555182 |
| Reference:
|
[M2] Mlček J.: Valuations of structures.Comment. Math. Univ. Carolinae (1979), 20 681-695. MR 0555183 |
| Reference:
|
[M3] Mlček J.: Monotonic valuations and valuations of triads of higher types.Comment. Math. Univ. Carolinae (1981), 22 377-398. MR 0620373 |
| Reference:
|
[V] Vopěnka P.: Mathematics in the Alternative Set Theory.Teubner Texte, Leipzig, 1979. MR 0581368 |
| . |
