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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)$\in C⇒\left(z,y,x\right)\notin C$ (asymmetry),

(x,y,z)$\in C⇒\left(y,z,x\right)\in C$ (cyclicity),

(x,y,z)$\in C$, (x,z,u)$\in C⇒\left(x,y,u\right)\in C$ (transitivity),

x,y,z$\in G$, $x\ne y\ne z\ne x⇒either$ (x,y,z)$\in C$ or (z,y,x)$\in C$ (linearity).

A cyclically ordered group is a group $\left(G,+\right)$ such that G is a cyclically ordered set and it holds $\left(x,y,z\right)\in C⇒\left(a+x+b,a+y+b,a+z+b\right)\in C$ for any a,b$\in G$. A cut on a cyclically ordered set G is a linear order $<$ on G such that $x. Such a cut is called regular if $\left(G,<\right)$ 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 $\left(G,+\right)$, $\left({G}_{1},{+}_{1}\right)$ be cyclically ordered groups such that ${G}_{1}\subseteq C\left(G\right)$ with the induced cyclic order and $\left(G,+\right)$ is a subgroup of $\left({G}_{1},{+}_{1}\right)$. Then $\left({G}_{1},{+}_{1}\right)$ is called an extension of $\left(G,+\right)$. The set of all extensions of $\left(G,+\right)$ is (partially) ordered by set inclusion; its greatest element is called a completion of $\left(G,+\right)$. 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 $\left(G,+\right)$, and a necessary and sufficient condition for a cyclically group $\left(G,+\right)$ to be complete i.e. equal to its completion.

Reviewer: V.Novák

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