Techreport,
Extendability, dimensions, and diagrams of cyclic orders.
Preprint, 944. Fachbereich Mathematik TH Darmstadt, Darmstadt, (October 1985)
DOI: 10.1137/0404041
Abstract
From the abstract and the introduction: Let $X$ be a set and $TX3$ be a set of triples. Then $C=(X,T)$ is called cyclic order iff the ternary relation $T$ is:1) cyclic, i.e., $(x,y,z)T$ implies $(y,z,x)T$;2) asymmetric, i.e., $(x,y,z)T$ implies $(z,y,x)\notT$;3) transitive, i.e., $(x,y,z),(x,z,w)T$ implies $(x,y,w)T$.A cyclic order $C=(X,T)$ is called a total cyclic order iff $T$ is:4) total, i.e., for each $x,y,zX$, $xyzx$, either $(x,y,z)T$ or $(z,y,x)T$.Several classes of cyclic orders arising from geometrical, algebraical, and combinatorial structures are introduced, and their extendability to total cyclic orders is studied. By analogy to Dushnik-Miller dimension for partial orders, for circular orders (i.e. for intersections of total cyclic orders) the intersection and product dimension (that may differ up to a factor of two) are defined. A class of cyclic orders that allow a graphic representation similar to Hasse diagrams is also studied.
