Some properties of Lorenzen ideal systems

Aleka Kalapodi; Angeliki Kontolatou; Jiří Močkoř

Archivum Mathematicum (2000)

  • Volume: 036, Issue: 4, page 287-295
  • ISSN: 0044-8753

Abstract

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Let G be a partially ordered abelian group ( p o -group). The construction of the Lorenzen ideal r a -system in G is investigated and the functorial properties of this construction with respect to the semigroup ( R ( G ) , ⊕ , ≤ ) of all r -ideal systems defined on G are derived, where for r , s ∈ R ( G ) and a lower bounded subset X ⊆ G , X r ⊕ s = X r ∩ X s . It is proved that Lorenzen construction is the natural transformation between two functors from the category of p o -groups with special morphisms into the category of abelian ordered semigroups.

How to cite

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Kalapodi, Aleka, Kontolatou, Angeliki, and Močkoř, Jiří. "Some properties of Lorenzen ideal systems." Archivum Mathematicum 036.4 (2000): 287-295. <http://eudml.org/doc/248571>.

@article{Kalapodi2000,
abstract = {Let $G$ be a partially ordered abelian group ($po$-group). The construction of the Lorenzen ideal $r_a$-system in $G$ is investigated and the functorial properties of this construction with respect to the semigroup $(R(G),\oplus ,\le )$ of all $r$-ideal systems defined on $G$ are derived, where for $r,s\in R(G)$ and a lower bounded subset $X\subseteq G$, $X_\{r\oplus s\}=X_r\cap X_s$. It is proved that Lorenzen construction is the natural transformation between two functors from the category of $po$-groups with special morphisms into the category of abelian ordered semigroups.},
author = {Kalapodi, Aleka, Kontolatou, Angeliki, Močkoř, Jiří},
journal = {Archivum Mathematicum},
keywords = {$r$-ideal; $r_a$-system; system of finite character; -ideal; Lorenzen ideal system; Lorenzen -group; system of finite character},
language = {eng},
number = {4},
pages = {287-295},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Some properties of Lorenzen ideal systems},
url = {http://eudml.org/doc/248571},
volume = {036},
year = {2000},
}

TY - JOUR
AU - Kalapodi, Aleka
AU - Kontolatou, Angeliki
AU - Močkoř, Jiří
TI - Some properties of Lorenzen ideal systems
JO - Archivum Mathematicum
PY - 2000
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 036
IS - 4
SP - 287
EP - 295
AB - Let $G$ be a partially ordered abelian group ($po$-group). The construction of the Lorenzen ideal $r_a$-system in $G$ is investigated and the functorial properties of this construction with respect to the semigroup $(R(G),\oplus ,\le )$ of all $r$-ideal systems defined on $G$ are derived, where for $r,s\in R(G)$ and a lower bounded subset $X\subseteq G$, $X_{r\oplus s}=X_r\cap X_s$. It is proved that Lorenzen construction is the natural transformation between two functors from the category of $po$-groups with special morphisms into the category of abelian ordered semigroups.
LA - eng
KW - $r$-ideal; $r_a$-system; system of finite character; -ideal; Lorenzen ideal system; Lorenzen -group; system of finite character
UR - http://eudml.org/doc/248571
ER -

References

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  6. Algebraic and categorical properties of r -ideal systems, International Journal of Mathematics and Mathematical Sciences, to appear. 
  7. Abstrakte Begründung der multiplikativen Idealtheorie, Math. Z. 45 (1939), 533–553. (1939) Zbl0021.38703MR0000604
  8. Groups of Divisibility, D. Reidl Publ. Co., Dordrecht, 1983. (1983) MR0720862
  9. Groups with quasi divisor theory, Comm. Math. Univ. St. Pauli, Tokyo 42 (1993), 23–36. (1993) MR1223185
  10. Divisorentheorie einer Halbgruppe, Math. Z. 114 (1970), 113–120. (1970) Zbl0177.03202MR0262401

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