Diversity in monoids

Jack Maney; Vadim Ponomarenko

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 3, page 795-809
  • ISSN: 0011-4642

Abstract

top
Let M be a (commutative cancellative) monoid. A nonunit element q ∈ M is called almost primary if for all a , b ∈ M , q ∣ a b implies that there exists k ∈ ℕ such that q ∣ a k or q ∣ 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.

How to cite

top

Maney, Jack, and Ponomarenko, Vadim. "Diversity in monoids." Czechoslovak Mathematical Journal 62.3 (2012): 795-809. <http://eudml.org/doc/247241>.

@article{Maney2012,
abstract = {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.},
author = {Maney, Jack, Ponomarenko, Vadim},
journal = {Czechoslovak Mathematical Journal},
keywords = {factorization; monoid; diversity; factorizations into almost primary elements; commutative cancellative monoids; diversity},
language = {eng},
number = {3},
pages = {795-809},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Diversity in monoids},
url = {http://eudml.org/doc/247241},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Maney, Jack
AU - Ponomarenko, Vadim
TI - Diversity in monoids
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 795
EP - 809
AB - 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.
LA - eng
KW - factorization; monoid; diversity; factorizations into almost primary elements; commutative cancellative monoids; diversity
UR - http://eudml.org/doc/247241
ER -

References

top
  1. Anderson, D. D., Mahaney, L. A., 10.1016/0022-4049(88)90026-6, J. Pure Appl. Algebra 54 (1988), 141-154. (1988) Zbl0665.13004MR0963540DOI10.1016/0022-4049(88)90026-6
  2. Geroldinger, A., 10.4064/cm-72-1-53-81, Colloq. Math. 72 (1997), 53-81. (1997) Zbl0874.13001MR1425546DOI10.4064/cm-72-1-53-81
  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). (2006) Zbl1113.11002MR2194494
  4. Geroldinger, A., Hassler, W., 10.1016/j.jpaa.2007.10.020, J. Pure Appl. Algebra 212 (2008), 1509-1524. (2008) Zbl1133.20047MR2391663DOI10.1016/j.jpaa.2007.10.020
  5. Halter-Koch, F., Divisor theories with primary elements and weakly Krull domains, Boll. Unione Mat. Ital., VII. Ser., B 9 (1995), 417-441. (1995) Zbl0849.20041MR1333970
  6. Halter-Koch, F., Ideal Systems. An Introduction to Multiplicative Ideal Theory, Pure and Applied Mathematics, Marcel Dekker, vol. 211, New York (1998). (1998) Zbl0953.13001MR1828371

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.