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Zbl 0856.06012
Jasem, Milan
Weak isometries and direct decompositions of partially ordered groups.
(English)
[J] Tatra Mt. Math. Publ. 5, 131-142 (1995). ISSN 1210-3195

Let $G$ be a partially ordered group. For $a\in G$, the set of all upper bounds for $\{a, - a\}$ is denoted by $|a|$. A mapping $f: G\to G$ is called a weak isometry if $|f(x)- f(y)|= |x- y|$ for all $x, y\in G$. $f$ is called stable when $f(0)= 0$. If $f: G\to G$ is a stable weak isometry, we denote $A_1= \{x\in G^+: f(x)= x\}$ and $B_1= \{x\in G^+: f(x)= - x\}$. This paper studies properties of $A:= A_1- A_1$ and $B:= B_1- B_1$. In the ideal situation, e.g., if $G$ is lattice-ordered, it follows that $G$ is the direct product of $A$ and $B$ and $f(x)= x(A)- x(B)$ for all $x\in G$. In an earlier paper, the author [Acta Math. Univ. Comen., New Ser. 63, 259-265 (1994; Zbl 0821.06016)] has shown that such a decomposition does not necessarily follow for every directed group and he investigated some conditions on stable weak isometries that are necessary and sufficient in order to obtain a direct composition of $G$.\par The author continues his investigations of properties of the sets $A$ and $B$ above and the relation between weak isometries and direct decompositions. He proves that the desired decomposition follows when $G$ is a Riesz group or when $G$ is an isolated Abelian directed group. [Reviewer's comment: There is a minor misprint in the definition of isolated group in the introduction of this paper.] Additionally, he obtains some results for Riesz groups. For instance, denoting $U(V)$ (respectively $L(V)$) to be the set of upper bounds (respectively lower bounds) of a subset $V$ of $G$, he derives the interesting formula $f(U(L(\{x, y\}))\cap L(U(\{x, y\})))=U(L(\{f(x), f(y)\}))\cap L(U(\{f(x), f(y)\}))$ (for all $x, y\in G$) in case $f$ is a weak isometry and $G$ has the Riesz interpolation property.
[G.Buskes (University/Mississippi)]
MSC 2000:
*06F15 Ordered groups

Keywords: partially ordered group; direct product; stable weak isometries; direct decompositions; Riesz group; Abelian directed group; isolated group; Riesz interpolation property

Citations: Zbl 0821.06016

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