Erweiterungen, Diagramme und Dimension zyklischer Ordnungen. (Extensions, diagrams and dimension of cyclic orders). (German) Zbl 0616.06001

Alles, Peter

Fachbereich Mathematik der Technischen Hochschule Darmstadt. 129 S. (1986).

This is a very good survey of the contemporary knowledge on cyclically ordered sets. A cyclic order on a set G is a ternary relation C on G which is asymmetric ((x,y,z)\(\in C\Rightarrow (z,y,x)\not\in C)\), cyclic ((x,y,z)\(\in C\Rightarrow (y,z,x)\in C)\) and transitive ((x,y,z)\(\in C\), (x,z,u)\(\in C\Rightarrow (x,y,u)\in C)\). If C is complete, i.e. x,y,z\(\in G\), \(x\neq y\neq z\neq x\Rightarrow\) either (x,y,z)\(\in C\) or (z,y,x)\(\in C\), then (G,C) is called a cycle or circle. The author introduces special kinds of cyclic orders as poset generated (G is a poset and (x,y,z)\(\in C\Leftrightarrow\) either \(x<y<z\) or \(y<z<x\) or \(z<x<y)\), arc-order (points are arcs on a circle and three arcs are in C iff they are pairwise disjoint and ordered clockwise), path-order (points are paths in a plane connecting two fixed points with a natural defined cyclic order) and others and finds relations between them. In 1976, N. Megiddo [Bull. Am. Math. Soc. 82, 274-276 (1976; Zbl 0361.06001)] has shown that not every cyclic order is embeddable into a circle. Here, a general construction is given for finding not totally extendable cyclic orders which are minimal with this property, i.e. omitting of any triple in C leads to a totally extendable one. A cyclic order is called circular iff it is an intersection of a family of circles. It is shown that circular orders are just substructures of direct products of circles and that substructures and products of circular orders are circular. For circular orders it is possible to define two kinds of dimension: intersection dimension idim C\(=\min \{| I|\); \(C=\cap_{i\in I}C_ i\), \(C_ i\) circle\(\}\), and product dimension pdim C\(=\min \{| I|\); \(C\subseteq \times_{i\in I}C_ i\), \(C_ i\) circle\(\}\). While for ordered sets these dimensions coincide, here it is shown pdim \(C\leq i\dim C\leq 2 p\dim C\) and both equalities hold in infinitely many cases. For some special cyclic orders, their intersection dimension is found. The paper ends with 14 unsolved problems.

Reviewer: V.Novák

MSC:

06A06	Partial orders, general
03E20	Other classical set theory (including functions, relations, and set algebra)

Keywords:

total extension; cyclically ordered sets; ternary relation; cyclic orders; circular orders; direct products of circles; intersection dimension; product dimension

Citations:

Zbl 0361.06001