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.


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


Zbl 0021.38703
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