| Title:
|
Diversity in monoids (English) |
| Author:
|
Maney, Jack |
| Author:
|
Ponomarenko, Vadim |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
3 |
| Year:
|
2012 |
| Pages:
|
795-809 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $M$ be a (commutative cancellative) monoid. A nonunit element $q\in M$ is called almost primary if for all $a,b\in M$, $q\mid ab$ implies that there exists $k\in \mathbb {N}$ such that $q\mid a^k$ or $q\mid b^k$. We introduce a new monoid invariant, diversity, which generalizes this almost primary property. This invariant is developed and contextualized with other monoid invariants. It naturally leads to two additional properties (homogeneity and strong homogeneity) that measure how far an almost primary element is from being primary. Finally, as an application the authors consider factorizations into almost primary elements, which generalizes the established notion of factorization into primary elements. (English) |
| Keyword:
|
factorization |
| Keyword:
|
monoid |
| Keyword:
|
diversity |
| MSC:
|
11B75 |
| MSC:
|
11N80 |
| MSC:
|
13A05 |
| MSC:
|
20M05 |
| MSC:
|
20M14 |
| idZBL:
|
Zbl 1265.20060 |
| idMR:
|
MR2984635 |
| DOI:
|
10.1007/s10587-012-0046-1 |
| . |
| Date available:
|
2012-11-10T21:18:01Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143026 |
| . |
| Reference:
|
[1] Anderson, D. D., Mahaney, L. A.: On primary factorizations.J. Pure Appl. Algebra 54 (1988), 141-154. Zbl 0665.13004, MR 0963540, 10.1016/0022-4049(88)90026-6 |
| Reference:
|
[2] Geroldinger, A.: Chains of factorizations in weakly Krull domains.Colloq. Math. 72 (1997), 53-81. Zbl 0874.13001, MR 1425546, 10.4064/cm-72-1-53-81 |
| Reference:
|
[3] Geroldinger, A., Halter-Koch, F.: Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory.Pure and Applied Mathematics (Boca Raton), vol. 278, Chapman & Hall/CRC, Boca Raton, FL (2006). Zbl 1113.11002, MR 2194494 |
| Reference:
|
[4] Geroldinger, A., Hassler, W.: Local tameness of {$v$}-{N}oetherian monoids.J. Pure Appl. Algebra 212 (2008), 1509-1524. Zbl 1133.20047, MR 2391663, 10.1016/j.jpaa.2007.10.020 |
| Reference:
|
[5] Halter-Koch, F.: Divisor theories with primary elements and weakly Krull domains.Boll. Unione Mat. Ital., VII. Ser., B 9 (1995), 417-441. Zbl 0849.20041, MR 1333970 |
| Reference:
|
[6] Halter-Koch, F.: Ideal Systems. An Introduction to Multiplicative Ideal Theory.Pure and Applied Mathematics, Marcel Dekker, vol. 211, New York (1998). Zbl 0953.13001, MR 1828371 |
| . |
