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Lexicographic products of cyclically ordered groups. (English) Zbl 0836.06010
In the present paper a lexicographic product of cyclically ordered groups (co-groups) is investigated. It is shown that two finite lexicographic product decompositions of isomorphic co-groups \({\mathfrak G}\) and \({\mathfrak H}\), \({\mathfrak G} = {\mathfrak G}_1 \circ {\mathfrak G}_2 \circ \dots \circ {\mathfrak G}_n\), \({\mathfrak H} = {\mathfrak H}_1 \circ {\mathfrak H}_2 \circ \dots \circ {\mathfrak H}_m\), have always isomorphic refinements provided that all \({\mathfrak G}_i\), \({\mathfrak H}_j\) \((i = 1,2, \dots, n\); \(j = 1,2, \dots, m)\) are lc-groups (see below for definitions of dc- and lc-), then \(m = n\), and \({\mathfrak G}_i\) is isomorphic with \({\mathfrak H}_i\) \((i = 1,2, \dots, n)\). Further, the cancellation in lexicographic product decompositions of co-groups is studied. Some definitions follow. An element \(x\) of a co-group \({\mathfrak G}\) is called isolated if there are no elements \(y,z \in G\) with the property \((x,y,z) \in C\). If every element of \({\mathfrak G}\) is isolated, then \({\mathfrak G}\) will be called isolated. We say that a co-group \({\mathfrak G}\) is an lc-group if \(\text{card} G > 2\), and if whenever \(x,y,z\) are distinct elements of \({\mathfrak G}\), then either \((x,y,z) \in C\) or \((z,y,x) \in C\). \({\mathfrak G}\) will be called a dc-group if for each \(x,y \in G\), \(x \neq y\), there exists an element \(z \in G\) such that either \((x,y,z) \in C\) or \((x,z,y) \in C\). Now, let \((G,+, \leq)\) be a partially ordered group. Define a ternary relation \(C_\leq\) on \(G\): For elements \(x,y,z \in G \) we put \((x,y,z) \in C_\leq\) iff \(x < y < z\) or \(y < z < x\) or \(z < x < y\). Then \((G,+,C_\leq)\) is a co- group. Assume that \(A\) and \(B\) are subgroups of a co-group \({\mathfrak G}\) such that the following conditions hold: (i) for each \(g\in G\) there exist uniquely determined elements \(a \in A\), \(b \in B\) such that \(g = a + b\), (ii) if \(g_i = a_i + b_i\), \(a_i \in A\), \(b_i \in B\), \((i = 1,2)\), then \(g_1 + g_2 = (a_1 + a_2) + (b_1 + b_2)\), (iii) if \(g_1, g_2, g_3\) are distinct elements of \(G\), \(g_i = a_i + b_i\) \((i = 1,2,3)\), then \((g_1, g_2, g_3) \in C\) iff either \((a_1, a_2, a_3) \in C\) or \(a_1 = a_2 = a_3\) and \((b_1, b_2, b_3) \in C\). Under these assumptions, we write \({\mathfrak G} = {\mathfrak A} \circ {\mathfrak B}\). This equation is called a lexicographic product decomposition of \({\mathfrak G}\) with factors \({\mathfrak A}\) and \({\mathfrak B}\). Similarly for \(n\) factors. Theorem. Let \((G,+, \leq)\) be a partially ordered (directed) group, and let \({\mathfrak G} = (G,+, C_\leq)\) be a co-group. If \((G,+, \leq) = {\mathfrak A} \circ {\mathfrak B}\), \(\text{card} A > 1\), \(\text{card} B > 1\), then \({\mathfrak G} = {\mathfrak A} \circ {\mathfrak B}\) iff either \({\mathfrak A}\) or \({\mathfrak B}\) is isolated \(({\mathfrak B}\) is isolated).

MSC:
06F15 Ordered groups
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