### Interval Orders and Reverse Mathematics

Alberto Marcone
Source: Notre Dame J. Formal Logic Volume 48, Number 3 (2007), 425-448.

#### Abstract

We study the reverse mathematics of interval orders. We establish the logical strength of the implications among various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order is an interval order if and only if it does not contain 2 \oplus 2. We also study proper interval orders and their characterization theorem: a partial order is a proper interval order if and only if it contains neither 2 \oplus 2 nor 3 \oplus 1.

First Page:
Primary Subjects: 03B30
Secondary Subjects: 06A06, 03D45
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1187031412
Digital Object Identifier: doi:10.1305/ndjfl/1187031412
Mathematical Reviews number (MathSciNet): MR2336356
Zentralblatt MATH identifier: 1135.03005

### References

[1] Balof, B., and K. Bogart, "Simple inductive proofs of the Fishburn and Mirkin theorem and the Scott-Suppes theorem", Order, vol. 20 (2003), pp. 49--51.
Mathematical Reviews (MathSciNet): MR1993409
Zentralblatt MATH: 1031.06001
Digital Object Identifier: doi:10.1023/A:1024430208672
[2] Bogart, K. P., and D. B. West, "A short proof that `proper = unit'", Discrete Mathematics, vol. 201 (1999), pp. 21--23.
Mathematical Reviews (MathSciNet): MR1687858
Zentralblatt MATH: 0932.05065
Digital Object Identifier: doi:10.1016/S0012-365X(98)00310-0
[3] Cenzer, D., and J. B. Remmel, "Proof-theoretic strength of the stable marriage theorem and other problems", pp. 67--103 in Reverse Mathematics 2001, vol. 21 of Lecture Notes in Logic, Association for Symbolic Logic, La Jolla, 2005.
Mathematical Reviews (MathSciNet): MR2185428
Zentralblatt MATH: 1087.03040
[4] Cholak, P., A. Marcone, and R. Solomon, "Reverse mathematics and the equivalence of definitions for well and better quasi-orders", The Journal of Symbolic Logic, vol. 69 (2004), pp. 683--712.
Mathematical Reviews (MathSciNet): MR2078917
Zentralblatt MATH: 1075.03030
Digital Object Identifier: doi:10.2178/jsl/1096901762
Project Euclid: euclid.jsl/1096901762
[5] Downey, R. G., D. R. Hirschfeldt, S. Lempp, and R. Solomon, "Computability-theoretic and proof-theoretic aspects of partial and linear orderings", Israel Journal of Mathematics, vol. 138 (2003), pp. 271--89.
Mathematical Reviews (MathSciNet): MR2031960
Zentralblatt MATH: 1044.03043
Digital Object Identifier: doi:10.1007/BF02783429
[6] Downey, R. G., and S. Lempp, "The proof-theoretic strength of the Dushnik-Miller theorem for countable linear orders", pp. 55--57 in Recursion Theory and Complexity (Kazan, 1997), vol. 2 of de Gruyter Series in Logic Applications, de Gruyter, Berlin, 1999.
Mathematical Reviews (MathSciNet): MR1724931
Zentralblatt MATH: 0951.03053
[7] Ershov, Yu. L., S. S. Goncharov, A. Nerode, J. B. Remmel, and V. W. Marek, editors, Handbook of Recursive Mathematics. Vol. 2. Recursive Algebra, Analysis and Combinatorics, vol. 139 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1998.
Mathematical Reviews (MathSciNet): MR1673582
Zentralblatt MATH: 0905.03002
[8] Fishburn, P. C., "Intransitive indifference with unequal indifference intervals", Journal of Mathematical Psychology, vol. 7 (1970), pp. 144--49.
Mathematical Reviews (MathSciNet): MR0253942
Zentralblatt MATH: 0191.31501
Digital Object Identifier: doi:10.1016/0022-2496(70)90062-3
[9] Fishburn, P. C., Interval Orders and Interval Graphs. A Study of Partially Ordered Sets, Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons Ltd., Chichester, 1985.
Mathematical Reviews (MathSciNet): MR776781
Zentralblatt MATH: 0551.06001
[10] Friedman, H. M., "Metamathematics of comparability", pp. 201--18 in Reverse Mathematics 2001, vol. 21 of Lecture Notes in Logic, Association for Symbolic Logic, La Jolla, 2005.
Mathematical Reviews (MathSciNet): MR2185435
Zentralblatt MATH: 1083.03051
[11] Hirst, J. L., "A survey of the reverse mathematics of ordinal arithmetic", pp. 222--34 in Reverse Mathematics 2001, vol. 21 of Lecture Notes in Logic, Association for Symbolic Logic, La Jolla, 2005.
Mathematical Reviews (MathSciNet): MR2185437
Zentralblatt MATH: 1088.03047
[12] Marcone, A., "Wqo and bqo theory in subsystems of second order arithmetic", pp. 303--30 in Reverse Mathematics 2001, vol. 21 of Lecture Notes in Logic, Association for Symbolic Logic, La Jolla, 2005.
Mathematical Reviews (MathSciNet): MR2185443
Zentralblatt MATH: 1083.03052
[13] Mirkin, B. G., "Description of some relations on the set of real-line intervals", Journal of Mathematical Psychology, vol. 9 (1972), pp. 243--52.
Mathematical Reviews (MathSciNet): MR0316345
Zentralblatt MATH: 0236.06002
Digital Object Identifier: doi:10.1016/0022-2496(72)90030-2
[14] Montalbán, A., "Equivalence between Fraï ssé's conjecture and Jullien's theorem", Annals of Pure and Applied Logic, vol. 139 (2006), pp. 1--42.
Mathematical Reviews (MathSciNet): MR2206250
Zentralblatt MATH: 1094.03045
Digital Object Identifier: doi:10.1016/j.apal.2005.03.001
[15] Pouzet, M., and N. Sauer, "From well-quasi-ordered sets to better-quasi-ordered sets", 31 pages, available at http://arxiv.org/abs/math.CO/0601119, 2006.
Mathematical Reviews (MathSciNet): MR2274316
[16] Roberts, F. S., "Indifference graphs", pp. 139--46 in Proof Techniques in Graph Theory (Proceedings of the Second Ann Arbor Graph Theory Conference, Ann Arbor, Michigan, 1968), Academic Press, New York, 1969.
Mathematical Reviews (MathSciNet): MR0252267
Zentralblatt MATH: 0193.24205
[17] Schmerl, J. H., "Reverse mathematics and graph coloring: Eliminating diagonalization", pp. 331--48 in Reverse Mathematics 2001, vol. 21 of Lecture Notes in Logic, Association for Symbolic Logic, La Jolla, 2005.
Mathematical Reviews (MathSciNet): MR2185444
Zentralblatt MATH: 1083.03053
[18] Schröder, B. S. W., Ordered Sets. An Introduction, Birkhäuser Boston Inc., Boston, 2003.
Mathematical Reviews (MathSciNet): MR1944415
Zentralblatt MATH: 1010.06001
[19] Scott, D., and P. Suppes, "Foundational aspects of theories of measurement", The Journal of Symbolic Logic, vol. 23 (1958), pp. 113--28.
Mathematical Reviews (MathSciNet): MR0115919
Zentralblatt MATH: 0084.24603
Digital Object Identifier: doi:10.2307/2964389
Project Euclid: euclid.jsl/1183733255
[20] Simpson, S. G., editor, Reverse Mathematics 2001, vol. 21 of Lecture Notes in Logic, Association for Symbolic Logic, La Jolla, 2005.
Mathematical Reviews (MathSciNet): MR2186912
Zentralblatt MATH: 1075.03002
[21] Simpson, S. G., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1999.
Mathematical Reviews (MathSciNet): MR1723993
Zentralblatt MATH: 0909.03048
[22] Trotter, W. T., "New perspectives on interval orders and interval graphs", pp. 237--86 in Surveys in Combinatorics, 1997 (London), vol. 241 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1997.
Mathematical Reviews (MathSciNet): MR1477749
Zentralblatt MATH: 0877.06001
[23] Wiener, N., "A contribution to the theory of relative position", Proceedings of the Cambridge Philosophy Society, vol. 17 (1914), pp. 441--49.
Zentralblatt MATH: 45.1150.10