## Some properties of Lorenzen ideal systems.(English)Zbl 1047.06011

Let $$G$$ be a directed po-group and $$X\mapsto X_r$$ be a mapping from the set of all lower bounded subsets $$X$$ of $$G$$ into the power set of $$G$$ fulfilling the following axioms: (1) $$X\subseteq X_r$$, (2) $$X\subseteq Y_r\implies X_r\subseteq Y_r$$, (3) $$\{a\}_r=a\cdot G^+=(a)$$ for all $$a\in G$$, (4) $$a\cdot X_r=(a\cdot X)_r$$ for all $$a\in G$$. This mapping is called $$r$$-system of ideals (in $$G$$) and was investigated in this form by P. Lorenzen [Math. Z. 45, 533–553 (1939; Zbl 0021.38703)]. The $$r$$-closed sets are called $$r$$-ideals and we can introduce for them in the natural way the multiplication $$\times _r$$. If we put for any lower bounded subset $$X$$ of $$G$$ $X_{r_a}=\bigcup _{K\subseteq X,\,K\text{ finite}}K_{r_a},$ where $$K_{r_a}=\{g\in G:\,g\cdot L_r\subseteq K_r\times _r L_r$$, for some finite $$L\subseteq G\}$$, then we get a new $$r$$-system of ideals $$r_a$$, which is called the Lorenzen ideal system. The operation $$\times _r$$ satisfies the cancellation law, and the quotient group of $$r$$-ideals $$\Lambda _r(G)$$ is called the Lorenzen $$r$$-group of $$G$$. This group is a lattice-ordered group which contains $$G$$ as an ordered subgroup.
Let $$R(G)$$ be the set of all $$r$$-systems on $$G$$ and for $$r,s\in R(G)$$ let $$r\leq s$$ mean $$X_s\subseteq X_r$$ and $$X_{r\oplus s}=X_r\cap X_s$$ for any lower bounded subset $$X$$ of $$G$$. In this paper the functorial properties of the construction of the Lorenzen ideal $$r_a$$-system in $$G$$ with respect to the ordered abelian semigroup $$(R(G),\oplus ,\leq )$$ are derived and it is shown that this construction is the natural transformation between two functors from the category of po-groups with special morphisms into the category of abelian ordered semigroups.

### MSC:

 06F15 Ordered groups 06F05 Ordered semigroups and monoids 18A23 Natural morphisms, dinatural morphisms

Zbl 0021.38703
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