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Lexicographic product decompositions of cyclically ordered groups. (English) Zbl 0952.06021
A group $$(G,+)$$ with a cyclic order $$C$$ is said to be cyclically ordered if $$(x_1,x_2,x_3)\in C$$ implies $$(a+x_1, a+x_2, a+x_3) \in C$$, $$(x_1+a, x_2 + a, x_3 +a) \in C$$ for any $$a\in G$$. It is referred to as a $$dc$$-group if for any $$x,y\in G$$ with $$x\not = y$$ there exists $$z\in G$$ such that either $$(x,y,z)\in C$$ or $$(y,x,z)\in C$$.
Let $$I$$ be a linearly ordered set, $$G_i$$ a $$dc$$-group with the cyclic ordering $$C_i$$ for any $$i\in I$$, $$G_0$$ the cartesian product of the groups $$G_i$$ $$(i\in I)$$. For any $$x=(x_i)_{i\in I} \in G_0$$ put $$I(x) = \{i\in I; x_i \not = 0\}$$. Let $$G$$ be the set of all $$x\in G_0$$ such that the set $$I(x)$$ is well-ordered. Then $$G$$ is a $$dc$$-group where the cyclic ordering $$C$$ is defined as follows: For $$x,y,z \in G$$ the condition $$(x,y,z)\in C$$ holds if and only if there exists $$i(1) \in I$$ such that $$(x_{i(1)},y_{i(1)},z_{i(1)})\in C_{i(1)}$$ and $$x_i = y_i = z_i$$ for any $$i\in I$$ with $$i<i(1)$$. This $$dc$$-group if denoted by $$[\Gamma _{i\in I} G_i]$$. If $$H$$ is a $$dc$$-group and $$\alpha$$ an isomorphism of $$H$$ onto $$[\Gamma _{i\in I} G_i]$$, then $$\alpha$$ is said to be a lexicographic product decomposition of $$H$$.
The author defines a refinement of a lexicographic product decomposition in a natural way and proves that any two lexicographic product decompositions of a $$dc$$-group have isomorphic refinements.
Reviewer: M.Novotný (Praha)

MSC:
 06F15 Ordered groups 20F60 Ordered groups (group-theoretic aspects)
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References:
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