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Cuts in cyclically ordered sets. (English) Zbl 0551.06002
A cyclic order \(C\) on a set \(G\) is a ternary relation which is asymmetric \(((x,y,z)\in C\) implies \((z,y,x)\in C)\), cyclic \(((x,y,z)\in C\) implies \((y,z,x)\in C)\), transitive ((x,y,z)\(\in C\), (x,z,u)\(\in C\) implies \((x,y,y)\in C)\) and linear (x,y,z\(\in G\), \(x\neq y\neq z\neq x\) implies \((x,y,z)\in C\) or \((z,y,x)\in C)\). The pair \((G,C)\) is called a cyclically ordered set. A cut on \((G,C)\) is a linear order \(<\) on \(G\) with the property: \(x<y<z\) implies \((x,y,z)\in C\). The paper is concerned with the study of properties of cuts. The connection between the dense sets defined by means of cuts and dense sets defined by means of linear orders is studied. Then, the set of all cuts on \((G,C)\) is structured as a cyclically ordered set and it is proved that it has no gaps. (The cut \((G,<)\) is a gap if it has neither a least nor a greatest element).
Reviewer: D.Lucanu

06A05 Total orders
06A06 Partial orders, general
Full Text: EuDML
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