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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$\in G$, $x\ne y\ne z\ne x\Rightarrow either$ (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 $(G,+)$, $(G\sb 1,+\sb 1)$ be cyclically ordered groups such that $G\sb 1\subseteq C(G)$ with the induced cyclic order and $(G,+)$ is a subgroup of $(G\sb 1,+\sb 1)$. Then $(G\sb 1,+\sb 1)$ 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

06F15Ordered groups
20F60Ordered groups (group aspects)
06B23Complete lattices, completions