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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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