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On determination of a cyclic order. (English) Zbl 0537.06004

A ternary relation C on a set G is called a cyclic order iff (i) \((x,y,z)\in C\) implies \((z,y,x)\not\in C,\) (ii) \((x,y,z)\in C\) and \((x,z,u)\in C\) imply \((x,y,u)\in C\) and (iii) \((x,y,z)\in C\) implies \((y,z,x)\in C.\) Let \((G,<)\) be an ordered set. Put \((x,y,z)\in C_<\) iff either \(x<y<z,\) or \(y<z<x\) or \(z<x<y\), then \(C_<\) is a cyclic order. Further, if a family \((<_ i)_{i\in I}\) of orderings on G satisfies certain conditions, then the union \(\cup_{i\in I}C_{<_ i}\) is a cyclic order on G. Any cyclic order on G has such representations. Theorems about the minimum cardinality w(G,C) of I of such representations of C, as well as the minimum cardinality W(G,C) of such I, when the orderings \(<_ i\) are all restricted as linear, are studied. For example, \(w(G,C)\leq card G.\) Another section is devoted to study a ternary relation on G, which is complementary in \(G^ 3\) to a cyclic order.
Reviewer: T.Ohkuma

MSC:

06A06 Partial orders, general
05A20 Combinatorial inequalities
20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
03E20 Other classical set theory (including functions, relations, and set algebra)
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References:

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