Partial cyclic order

In mathematics, a partial cyclic order is a ternary relation that generalizes a cyclic order in the same way that a partial order generalizes a linear order.

Definition

Over a given set, a partial cyclic order is a ternary relation $R$ that is:

• cyclic, i.e. it is invariant under a cyclic permutation: $R(x,y,z)\Rightarrow R(y,z,x)$
• asymmetric: $R(x,y,z)\Rightarrow \not R(z,y,x)$
• transitive: $R(x,y,z)$ and $R(x,z,u)\Rightarrow R(x,y,u)$

Constructions

Various constructions of partial cyclic orders have been studied, analogous to constructions of other mathematical structures. These include the direct sum, the direct product, the power, and the Dedekind–MacNeille completion of a set.

Extensions

The relationship between partial and total cyclic orders is more complex than the relationship between partial and total linear orders. To begin with, not every partial cyclic order can be extended to a total cyclic order. An example is the following relation on the first thirteen letters of the alphabet: {acd, bde, cef, dfg, egh, fha, gac, hcb} ∪ {abi, cij, bjk, ikl, jlm, kma, lab, mbc}. This relation is a partial cyclic order, but it cannot be extended with either abc or cba; either attempt would result in a contradiction.

The above was a relatively mild example. One can also construct partial cyclic orders with higher-order obstructions such that, for example, any 15 triples can be added but the 16th cannot. In fact, determining whether a general ternary relation (or indeed a partial cyclic order) extends to a total cyclic order is NP-complete, since it solves 3SAT. This is in stark contrast with the problem to recognize when a binary relation extends to a linear order, which can be solved in linear time. (In particular, every poset has a linear extension, even if it is infinite, by the Szpilrajn extension theorem.)

Several authors have considered the question of how complicated a non-extendable partial cyclic order must be. It is known that any partial cyclic order on 6 elements or fewer does extend to a total order. There is a single example (up to isomorphism) on 7 elements, which is constructed by adjoining an extra "point at infinity" to the standard example poset of dimension 3.

References

• Fiorini, Samuel; Fishburn, Peter C. (1 June 2003). "Extendability of Cyclic Orders". Order. 20 (2): 151–171. doi:10.1023/B:ORDE.0000009252.21331.22. ISSN 1572-9273. Retrieved 28 May 2026.
• Galil, Zvi; Megiddo, Nimrod (October 1977), "Cyclic ordering is NP-complete" (PDF), Theoretical Computer Science, 5 (2): 179–182, doi:10.1016/0304-3975(77)90005-6, retrieved 30 April 2011
• Megiddo, Nimrod (March 1976), "Partial and complete cyclic orders" (PDF), Bulletin of the American Mathematical Society, 82 (2): 274–276, doi:10.1090/S0002-9904-1976-14020-7, retrieved 30 April 2011
• Novák, Vítězslav (1982), "Cyclically ordered sets" (PDF), Czechoslovak Mathematical Journal, 32 (3): 460–473, doi:10.21136/CMJ.1982.101821, hdl:10338.dmlcz/101821, retrieved 30 April 2011
• Novák, Vítězslav; Novotný, Miroslav (1984a), "On a power of cyclically ordered sets" (PDF), Časopis Pro Pěstování Matematiky, 109 (4): 421–424, doi:10.21136/CPM.1984.118209, hdl:10338.dmlcz/118209, retrieved 30 April 2011
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Further reading

• Alles, Peter; Nešetřil, Jaroslav; Poljak, Svatopluk (1991), "Extendability, Dimensions, and Diagrams of Cyclic Orders", SIAM Journal on Discrete Mathematics, 4 (4): 453–471, doi:10.1137/0404041
• Bandelt, Hans–Jürgen; Chepoi, Victor; Eppstein, David (2010), "Combinatorics and geometry of finite and infinite squaregraphs" (PDF), SIAM Journal on Discrete Mathematics, 24 (4): 1399–1440, arXiv:0905.4537, doi:10.1137/090760301, S2CID 10788524, retrieved 23 May 2011
• Chajda, Ivan; Novák, Vítězslav (1985), "On extensions of cyclic orders" (PDF), Časopis Pro Pěstování Matematiky, 110 (2): 116–121, doi:10.21136/CPM.1985.108597, hdl:10338.dmlcz/108597, retrieved 30 April 2011
• Fishburn, P. C.; Woodall, D. R. (June 1999), "Cycle Orders", Order, 16 (2): 149–164, doi:10.1023/A:1006381208272, S2CID 37680085
• Haar, Stefan (2001), "Cyclic and partial order models for concurrency" (PDF), Geometry and Topology in Concurrency Theory GETCO '01, pp. 51–62, retrieved 23 May 2011
• Ille, Pierre; Ruet, Paul (30 April 2008), "Cyclic Extensions of Order Varieties", Electronic Notes in Theoretical Computer Science, 212: 119–132, CiteSeerX 10.1.1.103.2305, doi:10.1016/j.entcs.2008.04.057 {{citation}}: Cite uses deprecated parameter |citeseerx= (help)
• Jakubík, Ján (1994), "On extended cyclic orders" (PDF), Czechoslovak Mathematical Journal, 44 (4): 661–675, doi:10.21136/CMJ.1994.128486, hdl:10338.dmlcz/128486, retrieved 30 April 2011
• Melliès, Paul-André (2004), "A topological correctness criterion for non-commutative logic" (PDF), in Thomas Ehrhard and Jean-Yves Girard and Paul Ruet and Philip Scott (ed.), Linear Logic in Computer Science, pp. 283–323, retrieved 23 May 2011
• Novák, Vítězslav (1984), "On some minimal problem" (PDF), Archivum Mathematicum, 20 (2): 95–99, hdl:10338.dmlcz/107191, MR 0784860, Zbl 0554.06003, retrieved 23 May 2011
• Stehr, Mark-Oliver (1998), "Thinking in Cycles", in Desel, Jörg; Silva, Manuel (eds.), ICATPN '98 Proceedings of the 19th International Conference on Application and Theory of Petri Nets, Lecture Notes in Computer Science, vol. 1420, pp. 205–225, doi:10.1007/3-540-69108-1_12, ISBN 3-540-64677-9
• Haar, Stefan (2016), "Cyclic Ordering through Partial Orders" (PDF), Journal of Multiple-Valued Logic and Soft Computing, 27 (2–3), Old City Publishing: 209–228