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On completion of cyclically ordered sets. (English) Zbl 0636.06004
A cyclic order on a set G is a ternary relation C on G, which is: (i) asymmetric, i.e. (x,y,z)\(\in C\) implies (z,y,x)\({\bar \in}C\), (ii) cyclic, i.e. (x,y,z)\(\in C\) implies (y,z,x)\(\in C\), and (iii) transitive, i.e. (x,y,z)\(\in C\) and (x,z,u)\(\in C\) together imply (x,y,u)\(\in C.\)
If, moreover, card \(G\geq 3\), and for any three pairwise different elements either (x,y,z)\(\in C\) or (z,y,x)\(\in C\) then C is linear, and (G,C) is called a linearly cyclically ordered set or a cycle. Earlier [Czech. Math. J. 34(109), 322-333 (1984; Zbl 0551.06002)] the first author has constructed a completion of cycles by means of cuts. In this note the authors give another construction of a completion, which can be applied to a larger class of monodimensional cyclically ordered sets.
Reviewer: S.Gacsályi

MSC:
06A06 Partial orders, general
06B23 Complete lattices, completions
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References:
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