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Dimension theory for cyclically and cocyclically ordered sets. (English) Zbl 0538.06002

Let C be a ternary relation on a set G. Consider the following 6 conditions on C:
(i) asymmetric - (x,y,z)\(\in C \Rightarrow\) (z,y,x)\(\not\in C\)
(ii) transitive - (x,y,z),(x,z,u)\(\in C \Rightarrow\) (x,y,u)\(\in C\)
(iii) cyclic - (x,y,z)\(\in C \Rightarrow\) (y,z,x)\(\in C\)
(iv) reflexive - car\(d\{\) x,y,\(z\} \leq 2 \Rightarrow\) (x,y,z)\(\in C\)
(v) complete - \(card\{x,y,z\}=3 \Rightarrow\) (x,y,z) or (z,y,x)\(\in C\)
(vi) For arbitrary \(u\in G\), (x,y,z)\(\in C \Rightarrow\) (x,y,u) or (x,u,z)\(\in C.\)
The pair (G,C) is called a cyclically ordered set in case C satisfies the first three conditions, and a cocyclically ordered set if conditions (iii)-(vi) hold. If \(\leq\) is a linear ordering of G, and if \(C_{<}\) is defined by \((x,y,z)\in C_{<}\) iff \(x<y<z\) or \(y<z<x\) or \(z<x<y,\) then \(C_{<}\) is a cyclic order. Every cyclic order is the union of a family of suitable \(C_{<}\)-orders. Cocyclic orders are characterized as being the complements (in \(G^ 3)\) of cyclic orders. The paper is concerned with the development of a dimension theory for cyclic and cocyclically ordered sets.
Reviewer: M.F.Janowitz

MSC:

06A06 Partial orders, general
20N10 Ternary systems (heaps, semiheaps, heapoids, etc.)
08A02 Relational systems, laws of composition
06A05 Total orders
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References:

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