Abstract
In this paper a result of B. Leclerc and B. Monjardet concerning meet-projections in finite congruence-simple atomistic lattices is generalized. We prove that the result remains valid for any finite tolerance-simple lattice; moreover, we extend it to a type of subdirect product of such lattices, introducing the notion of a generalized oligarchy.
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Arrow, K.J.: Social Choice and Individual Values. Wiley, New York (1951)
Blyth, T.S., Janowitz, M.F.: Residuation Theory. Pergamon (1972)
Chajda, I.: Algebraic Theory of Tolerance Relations. University Palackého, Olomouc (1991)
Chambers, C.P., Miller, A.D.: Rules for aggregating information. Soc. Choice Welf. 36, 75–82 (2011)
Day, A.: Characterization of finite lattices that are bounded homomorphic images or sublattices of free lattices. Can. J. Math. 31, 69–78 (1979)
Day, W.H.E., McMorris, F.R.: Axiomatic consensus theory in group choice and biomathematics. SIAM Frontiers in Mathematics, vol. 29. Siam, Philadelphia PA (2003)
Dilworth, R.P.: The structure of relatively complemented lattices. Ann. Math. 51(2), 348–359 (1950)
Dimitrov, D., Marchant, T., Mishra, N.: Separability and aggregation of equivalence relations. Econ. Theory 51(1), 191–212 (2012)
Erné, M.: Adjunctions and Galois connections: origins, history and development. In: Denecke, K., Erné, M., Wismath, S.L. (eds.) Galois Connections and Applications, vol. 565, pp. 1–138. Springer Science and Business Media (2013)
Erné, M., Reinhold, J.: Intervals in lattices of quasiorders. Order 12(4), 375–403 (1995)
Freese, R., Jezek, K., Nation, J.B.: Free lattices. American Mathematics Social, Providence (1995)
Ganter, B., Wille, R.: Formal Concept Analysis, Mathematical Foundations. Springer, Berlin (1999)
Ganter, B., Körei, A., Radeleczki, S.: Extent partitions and context extensions. Math. Slovaka 63(4), 693–706 (2013)
Grätzer, G.: Lattice Theory: Foundation. Birkhäuser (2010)
Jakubíková-Studenovská, D., Pöschel, R., Radeleczki, S.: The lattice of quasiorder lattices of algebras on a finite set. Algebra Universalis. in print
Janowitz, M.F.: Decreasing Baer semigroups. Glasgow. Math. J. 10, 46–51 (1969)
Janowitz, M.F.: Tolerances and congruences. Czechoslov. Math. J 36, 108–115 (1986)
Janowitz, M.F.: Tolerances, interval orders and semiorders. Czechoslov. Math. J. 44, 21–37 (1994)
Janowitz, M.F.: Generalized oligarchies. Presented to the American Mathematics Society, October 5. (Meeting 1092) (2013)
Janowitz, M.F.: Some observations on oligarchies, internal direct sums and lattice congruences. In: Pardalos, P., Goldengorin, B., Alekserov, F. (eds.) Clusters, Orders, Trees: Methods and applications, pp. 231–247. Springer, New York (2014)
Leclerc, B.: Consensus applications in the social sciences. In: Bock, H.H. (ed.) Classification and Related Methods of Data Analysis, pp. 333–340. North Holland, Amsterdam (1988)
Leclerc, B., Monjardet, B.: Latticial theory of consensus. In: Barnett, V., Molin, M., Salles, M., Schofield, N. (eds.) Social Choice, Welfare and Ethics, pp. 145–160. Cambridge University Press (1995)
Leclerc, B., Monjardet, B.: Aggregation and residuation. Order 30, 261–268 (2013)
Mirkin, B.G.: On the problem of reconciling partitions. In: Quantitative Sociology, International Perspective on Mathematical and Statistical Modelling, pp. 441–449. Academic Press, New York (1975)
Monjardet, B.: Arrowian characterization of latticial federation consensus functions. Math. Soc. Sci. 20, 51–71 (1990)
Monjardet, B., Caspard, N.: On a dependence relation in finite lattices. Discr. Math. 165/166, 497–505 (1997)
Radeleczki, S.: Classification systems and the decomposition of a lattice into direct products. Math. Notes Miskolc. 1, 145–156 (2000)
Radeleczki, S.: Classification systems and their lattice. Discussiones Mathematicae, Gen. Algebra Appl. 22, 167–181 (2002)
Szász, G.: Introduction to Lattice theory, 3rd rev. ed. Akad. Kiad. Budapest. Academic Press, New York (1964)
Tu̇ma, J.: On the structure of quasi-ordering lattices. Acta Universitatis Carolinae, Mathematica et Physica 43(1), 65–74 (2002)
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Janowitz, M.F., Radeleczki, S. Aggregation on a Finite Lattice. Order 33, 371–388 (2016). https://doi.org/10.1007/s11083-015-9373-9
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DOI: https://doi.org/10.1007/s11083-015-9373-9