Some remarks on Lorenzen \(r\)-group of partly ordered groups. (English) Zbl 0879.20031

Močkoř, Jiří; Kontolatou, Angeliki

Czech. Math. J. 46, No. 3, 537-552 (1996).

A lot of problems concerning arithmetics of integral domains lead to the ideal theory of a ring and even to the abstract ideal theory of a semigroup. The investigation of these questions is involved in the theory of partially ordered groups. One of the most important notions of this theory, an \(r\)-system of ideals investigated by P. Lorenzen [Math. Z. 45, 533-553 (1939; Zbl 0021.38703)], is defined as follows.
Let \(G\) be a directed partially ordered commutative group. An \(r\)-system of ideals in \(G\) is a map \(X\to X_r\) from the set of all lower bounded (or only finite) subsets of \(G\) satisfying the following conditions: (1) \(X\subseteq X_r\), (2) \(X\subseteq Y_r\Rightarrow X_r\subseteq Y_r\), (3) \(\{a\}_r=a\cdot G^+=(a)\) for all \(a\in G\), (4) \(a\cdot X_r=(a\cdot X)_r\) for all \(a\in G\).
The set \(X_r\) is called an \(r\)-ideal of \(G\). The inclusion relation on the set of all \(r\)-ideals forms an order and the operation \(\times\) on this set is defined by \(X_r\times Y_r=(X_r\cdot Y_r)_r=(X\cdot Y)_r\). If \(\{x\in G:x\cdot X_r\subseteq X_r\}\subseteq G^+\) for each \(r\)-ideal \(X_r\) (\(X\) is finite), then \(G\) is called \(r\)-closed. In this case another \(r\)-system \(r_a\) in \(G\) is defined by \(A_{r_a}=\{g\in G:g\cdot K_r\subseteq A_r\times K_r,\text{ for some finite }K\subset G\}\) for each finite subset \(A\) of \(G\). The semigroup of all \(r_a\)-ideals with the operation \(\times\) satisfies the cancellation law and its quotient group \(\Lambda_r(G)\) is called the Lorenzen \(r\)-group of \(G\). The group \(\Lambda_r(G)\) forms a lattice ordered group if \(\Lambda_r(G)^+=\{A_{r_a}/B_{r_a}:A_{r_a}\subseteq B_{r_a}\}\).
The goal of this article is to study similarities of some properties of directed po-groups and their Lorenzen groups. The relationships between the structure of o-ideals in a directed po-group and some \(\ell\)-ideals of its Lorenzen \(r\)-group are investigated. A great part of this paper is devoted to the po-groups possessing a theory of quasi-divisors (generalizing the notion of the theory of divisors). If the quasi-divisors have a finite character, then for a special \(r\)-system of ideals the Lorenzen group is isomorphic to a special \(\ell\)-group of “compatible elements” of the product \(\prod_{w\in W}G_w\), where \(W\) is a defining family of certain valuations (\(t\)-valuations) on \(G\).
Finally, the authors show that a po-group \(G\) with a theory of quasi-divisors and its Lorenzen group \(\Lambda_r(G)\) have analogous properties in case the \(v\)-system and \(t\)-system are equal. (The systems \(v\) and \(t\) are defined in the usual way and an \(r\)-system is given).

Reviewer: L.Skula (Brno)

MSC:

20M12	Ideal theory for semigroups
20F60	Ordered groups (group-theoretic aspects)
13A05	Divisibility and factorizations in commutative rings
06F20	Ordered abelian groups, Riesz groups, ordered linear spaces
06F15	Ordered groups
20M14	Commutative semigroups
13A15	Ideals and multiplicative ideal theory in commutative rings

Keywords:

\(r\)-systems of ideals in po-groups; Lorenzen \(r\)-groups of po-groups; theory of quasi-divisors

Citations:

Zbl 0021.38703

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