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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

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