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Completion of a cyclically ordered group. (English) Zbl 0624.06021
A cyclically ordered set is a set G with a ternary relation C which satisfies
(x,y,z)$$\in C\Rightarrow (z,y,x)\not\in C$$ (asymmetry),
(x,y,z)$$\in C\Rightarrow (y,z,x)\in C$$ (cyclicity),
(x,y,z)$$\in C$$, (x,z,u)$$\in C\Rightarrow (x,y,u)\in C$$ (transitivity),
x,y,z$$\in G$$, $$x\neq y\neq z\neq x\Rightarrow either$$ (x,y,z)$$\in C$$ or (z,y,x)$$\in C$$ (linearity).
A cyclically ordered group is a group $$(G,+)$$ such that G is a cyclically ordered set and it holds $$(x,y,z)\in C\Rightarrow (a+x+b,a+y+b,a+z+b)\in C$$ for any a,b$$\in G$$. A cut on a cyclically ordered set G is a linear order $$<$$ on G such that $$x<y<z\Rightarrow (x,y,z)\in C$$. Such a cut is called regular if $$(G,<)$$ either contains a least element or has neither a least nor a greatest element. The set C(G) of all regular cuts on G with naturally defined cyclic order is called a completion of G.
Let $$(G,+)$$, $$(G_ 1,+_ 1)$$ be cyclically ordered groups such that $$G_ 1\subseteq C(G)$$ with the induced cyclic order and $$(G,+)$$ is a subgroup of $$(G_ 1,+_ 1)$$. Then $$(G_ 1,+_ 1)$$ is called an extension of $$(G,+)$$. The set of all extensions of $$(G,+)$$ is (partially) ordered by set inclusion; its greatest element is called a completion of $$(G,+)$$. The authors give a (challenging) construction of a completion of a cyclically ordered group. Also, they derive necessary and sufficient conditions under which a given cut belongs to a completion of a cyclically ordered group $$(G,+)$$, and a necessary and sufficient condition for a cyclically group $$(G,+)$$ to be complete i.e. equal to its completion.
Reviewer: V.Novák

##### MSC:
 06F15 Ordered groups 20F60 Ordered groups (group-theoretic aspects) 06B23 Complete lattices, completions
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