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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)

06F15 Ordered groups
20F60 Ordered groups (group-theoretic aspects)
Full Text: DOI EuDML
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