# zbMATH — the first resource for mathematics

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

##### MSC:
 06A05 Total orders 06A06 Partial orders, general
##### Keywords:
cyclically ordered set; linear order; cuts; dense sets
Full Text:
##### References:
 [1] Birkhoff G.: Generalized arithmetic. Duke Math. Journ. 9 (1942), 283-302. · Zbl 0060.12609 · doi:10.1215/S0012-7094-42-00921-9 [2] Čech E.: Bodové množiny. (Point Sets). Academia Praha, 1966. [3] Müller G.: Lineare und zyklische Ordnung. Praxis Math. 16 (1974), 261 - 269. · Zbl 0369.06001 [4] Novák V.: Cyclically ordered sets. Czech. Math. Journ. 32 (107) (1982), 460-473. · Zbl 0515.06003 · eudml:13330
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.