The fixed point and the common fixed point properties in finite pseudoordered sets.
Keywords:
Pseudoordered set, Trellis, Complete trellis, Monotone map, Fixed point property, Least fixed point property, Greatest fixed point property, Common fixed point propertyAbstract
In this paper, we first prove that every finite nonempty pseudoordered with a least element has the least fixed point property and the least common fixed point property for every finite commutative family of self monotone maps. Dually, we establish that a finite nonempty pseudoordered with a greatest element has the greatest fixed point property and the greatest common fixed point property for every finite commutative family of self monotone maps. Secondly, we prove that every monotone map ƒ defined on a nonempty finite pseudoordered (X, ⊵) has at least a fixed point if and only if there is at least an element ɑ of X such that the subset of X defined by {ƒn(ɑ) : n ∈ ℕ } has a least or a greatest element. Furthermore, we show that the set of all common fixed points of every finite commutative family of monotone maps defined on a finite nonempty complete trellis is also a nonempty complete trellis.
References
Abian, A., Fixed point theorems of the mappings of partially ordered sets, Rendiconti del Circolo Mathematico di Palermo 20, pp. 139142, (1971).
Abian, S. and Brown, A. B., A theorem on partially ordered sets with applications to fixed point theorems, Can. J. Math. 13, pp. 7882, (1961).
Amann, H., Order structures and fixed points, Ruhruniversităt, Bochum, mimeographed lecture notes, (1977).
Baclawski, K., A combinatorial proof of a fixed point property, J. Combin. Theory (A) 119, pp. 9941013, (2012).
Bourbaki, N., Sur le thorme de Zorn, Arch. Math., 2, pp. 434437, (19401950).
Demarr, R., Common fixed points for isotone mappings, Colloq. Math. 13, pp. 4548, (1964).
Bhatta, S. P. and Shashirekha, H., A characterisation of completeness for Trellises, Algebra univeralis 44, pp. 305308, (2000).
Bhatta, S. P., Weak chain completeness and fixed point property for pseudoordered sets, Czechoslovac Mathematical Journal, 55 (130), pp. 365369, (2005).
Bhatta, S. P. and Shiju, G., A note on weak chaincompleteness and fixed point property for pseudo ordered sets, Advances in discrete mathematics and applications: Mysore, 2008, 119123, Ramanujan Math. Soc. Lect. Notes Ser. 13, Ramanujan Math. Soc., Mysore, (2010).
Bhatta, S. P. and Shiju, G., Some fixed point theorems for pseudo ordered sets, Algebra Discrete Math. 11, No. 1, pp. 1722, (2011).
Brondsted, A., Common fixed points and partial orders, Proc. Amer. Math. Soc. 77, No. 3, pp. 365368, (1979).
Capard, N., Leclerc, B. and Monjardet B., Finite ordered sets, Concepts, results and uses. Encyclopedia of Mathematics and its Applications, 144. Cambridge University Press, Cambridge, (2012). xii+337 pp.
Knaster, B., Un théorème sur les fonctions d’ensembles, Ann. Soc. Polon. Math. 6 (1928), 133134.
Lim, T. C., On the largest common fixed point of a commuting family of isotone maps, Discrete and Continuous Dynamical Systems, Series A, Suppl., pp. 621623, (2005).
Rival, I., A fixed point theorem for finite partially ordered sets. J. Combin. Theory (A) 21(1976), 309318 (1976).
Schröder, Bernd S. W., Algorithms for the fixed point property, ORDAL ’96 (Ottawa, ON), Theoret. Comput. Sci. 217, No. 2, pp. 301358, (1999).
Schröder, Bernd S. W., Ordered sets. An introduction, Birkhuser Boston, Inc., Boston, MA, (2003). xviii+391 pp.
Schröder, Bernd S. W., The fixed point property for ordered sets. Arab. J. Math. (Springer) 1, No. 4, pp. 529547, (2012).
Skala, H. L., Trellis theory, Algebra Universalis 1, pp. 218233, (1971).
Skala, H. L., Trellis theory, Mem. Amer. Math. Soc. 121, Providence, (1972).
Stouti, A. and Maaden, A., Fixed points and common fixed points theorems in pseudoordered sets, Proyecciones 32, No. 4, pp. 409418, (2013).
Tarski, A., A latticetheoretical fixpoint theorem and its applications, Proc. Amer. Math. Soc. 5, pp. 285309, (1955).
Wong, James S. W., Common fixed points of commuting monotone mappings, Canad. J. Math., 19, pp. 617620, (1967).
Published
How to Cite
Issue
Section

Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
 No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.