Abstract
We investigate the structure of acyclic binary relations from different points of view. On the one hand, given a nonempty set we study real-valued bivariate maps that satisfy suitable functional equations, in a way that their associated binary relation is acyclic. On the other hand, we consider acyclic directed graphs as well as their representation by means of incidence matrices. Acyclic binary relations can be extended to the asymmetric part of a linear order, so that, in particular, any directed acyclic graph has a topological sorting.
This work has been partially supported by the research projects MTM2012-37894-C02-02, TIN2013-47605-P, ECO2015-65031-R, MTM2015-63608-P (MINECO/FEDER), TIN2016-77356-P and the Research Services of the Public University of Navarre (Spain).
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Alcantud, J.C.R.: Weak utilities from acyclicity. Theor. Decis. 47(2), 185–196 (1999)
Beardon, A.F., Candeal, J.C., Herden, G., Induráin, E., Mehta, G.B.: The non-existence of a utility function and the structure of non-representable preference relations. J. Math. Econom. 37, 17–38 (2002)
Bridges, D.S., Mehta, G.B.: Representations of Preference Orderings. Springer, Berlin (1995). https://doi.org/10.1007/978-3-642-51495-1
Campión, M.J., Catalán, R.G., Induráin, E., Ochoa, G.: Reinterpreting a fuzzy subset by means of a Sincov’s functional equation. J. Intell. Fuzzy Syst. 27, 367–375 (2014)
Hansson, B.: Choice structures and preference relations. Synthese 18, 443–458 (1968)
Kahn, A.B.: Topological sorting of large networks. Commun. ACM 5(11), 558–562 (1962)
Knuth, D.E., Szwarcfiter, J.L.: A structured program to generate all topological sorting arrangements. Inform. Process. Lett. 2(6), 153–157 (1974)
Kruskal, J.B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Amer. Math. Soc. 7, 48–50 (1956)
Rodríguez-Palmero, C.: A representation of acyclic preferences. Econom. Lett. 54, 143–146 (1997)
Sincov, D.M.: Über eine Funktionalgleichung. Arch. Math. Phys. 6(3), 216–227 (1903)
Suzumura, K.: Remarks on the theory of collective choice. Economica 43(172), 381–390 (1976)
Szpilrajn, E.: Sur l’ extension de l’ ordre partiel. Fund. Math. 16, 386–389 (1930)
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Alcantud, J.C.R., Campión, M.J., Candeal, J.C., Catalán, R.G., Induráin, E. (2018). On the Structure of Acyclic Binary Relations. In: Medina, J., Ojeda-Aciego, M., Verdegay, J., Perfilieva, I., Bouchon-Meunier, B., Yager, R. (eds) Information Processing and Management of Uncertainty in Knowledge-Based Systems. Applications. IPMU 2018. Communications in Computer and Information Science, vol 855. Springer, Cham. https://doi.org/10.1007/978-3-319-91479-4_1
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DOI: https://doi.org/10.1007/978-3-319-91479-4_1
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