# Article

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Keywords:
$r$-ideal; $r_a$-system; system of finite character
Summary:
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.
References:
[1] Arnold, I.: Ideale in kommutativen Halbgruppen. Rec. Math. Soc. Math. Moscow 36 (1929), 401–407.
[2] Borewicz, I. R., Shafarevicz, Z. I.: Number Theory. Academic Press, New York, 1966. MR 0195803
[3] Clifford, A. H.: Arithmetic and ideal theory of abstract multiplication. Ann. of Math. 39 (1938), 594–610. MR 1503427
[4] Halter-Koch, F.: Ideal systems. Marcel Dekker, Inc, New York - Basel - Hong Hong, 1998. MR 1828371 | Zbl 0953.13001
[5] Jaffard, P.: Les systémes d’idéaux. Dunod, Paris, 1960. MR 0114810 | Zbl 0101.27502
[6] Kalapodi, A. and Kontolatou, A.: Algebraic and categorical properties of $r$-ideal systems. International Journal of Mathematics and Mathematical Sciences, to appear.
[7] Lorenzen, P.: Abstrakte Begründung der multiplikativen Idealtheorie. Math. Z. 45 (1939), 533–553. MR 0000604 | Zbl 0021.38703
[8] Močkoř, J.: Groups of Divisibility. D. Reidl Publ. Co., Dordrecht, 1983. MR 0720862
[9] Močkoř, J., Kontolatou, A.: Groups with quasi divisor theory. Comm. Math. Univ. St. Pauli, Tokyo 42 (1993), 23–36. MR 1223185
[10] Skula, L.: Divisorentheorie einer Halbgruppe. Math. Z. 114 (1970), 113–120. MR 0262401 | Zbl 0177.03202

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