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Zbl 0836.06010
Černák, Štefan
Lexicographic products of cyclically ordered groups. (English)
[J] Math. Slovaca 45, No.1, 29-38 (1995). ISSN 0139-9918; ISSN 1337-2211/e

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 ${\germ G}$ and ${\germ H}$, ${\germ G} = {\germ G}_1 \circ {\germ G}_2 \circ \dots \circ {\germ G}_n$, ${\germ H} = {\germ H}_1 \circ {\germ H}_2 \circ \dots \circ {\germ H}_m$, have always isomorphic refinements provided that all ${\germ G}_i$, ${\germ H}_j$ $(i = 1,2, \dots, n$; $j = 1,2, \dots, m)$ are lc-groups (see below for definitions of dc- and lc-), then $m = n$, and ${\germ G}_i$ is isomorphic with ${\germ H}_i$ $(i = 1,2, \dots, n)$. Further, the cancellation in lexicographic product decompositions of co-groups is studied. Some definitions follow. An element $x$ of a co-group ${\germ G}$ is called isolated if there are no elements $y,z \in G$ with the property $(x,y,z) \in C$. If every element of ${\germ G}$ is isolated, then ${\germ G}$ will be called isolated. We say that a co-group ${\germ G}$ is an lc-group if $\text {card} G > 2$, and if whenever $x,y,z$ are distinct elements of ${\germ G}$, then either $(x,y,z) \in C$ or $(z,y,x) \in C$. ${\germ G}$ will be called a dc-group if for each $x,y \in G$, $x \ne y$, there exists an element $z \in G$ such that either $(x,y,z) \in C$ or $(x,z,y) \in C$. Now, let $(G,+, \le)$ be a partially ordered group. Define a ternary relation $C_\le$ on $G$: For elements $x,y,z \in G $ we put $(x,y,z) \in C_\le$ iff $x < y < z$ or $y < z < x$ or $z < x < y$. Then $(G,+,C_\le)$ is a co- group. Assume that $A$ and $B$ are subgroups of a co-group ${\germ G}$ 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 ${\germ G} = {\germ A} \circ {\germ B}$. This equation is called a lexicographic product decomposition of ${\germ G}$ with factors ${\germ A}$ and ${\germ B}$. Similarly for $n$ factors. Theorem. Let $(G,+, \le)$ be a partially ordered (directed) group, and let ${\germ G} = (G,+, C_\le)$ be a co-group. If $(G,+, \le) = {\germ A} \circ {\germ B}$, $\text {card} A > 1$, $\text {card} B > 1$, then ${\germ G} = {\germ A} \circ {\germ B}$ iff either ${\germ A}$ or ${\germ B}$ is isolated $({\germ B}$ is isolated).
[J.Ševečková (Brno)]

MSC 2000:: *06F15 Ordered groups

Keywords: refinement; lexicographic product; cyclically ordered groups; cancellation; lexicographic product decomposition

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