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Abstrakty
In this paper, we consider tolerances induced by irredundant coverings. Each tolerance R on U determines a quasiorder .≤R by setting x .≤R y if and only if R(x) ⊆ R(y). We prove that for a tolerance R induced by a covering H of U, the covering H is irredundant if and only if the quasiordered set (U,.≤R ) is bounded by minimal elements and the tolerance R coincides with the product .≤R ◦ .≤R . We also show that in such a case H = {↑m | m is minimal in (U,.≤R )}, and for each minimal m, we have R(m) = ↑m. Additionally, this irredundant covering H inducing R consists of some blocks of the tolerance R. We give necessary and sufficient conditions under which H and the set of R-blocks coincide. These results are established by applying the notion of Helly numbers of quasiordered sets.
Wydawca
Czasopismo
Rocznik
Tom
Strony
341--353
Opis fizyczny
Bibliogr. 19 poz., rys.
Twórcy
autor
- Sirkankuja 1, 20810 Turku, Finland
autor
- Institute of Mathematics, University of Miskolc 3515 Miskolc-Egyetemv´aros, Hungary
Bibliografia
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- [2] W. Bartol, J. Mir´o, K. Pi´oro, and F. Rossell´o, On the coverings by tolerance classes, Information Sciences 166 (2004), 193–211.
- [3] I. Chajda, Algebraic Theory of Tolerance Relations, Monograph series of Palackỷ University, 1991.
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- [9] F. Harary and I. C. Ross, A procedure for clique detection using the group matrix, Sociometry 20 (1957), 205–216.
- [10] J. Järvinen, Lattice theory for rough sets, Transactions on Rough Sets VI (2007), 400–498.
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- [12] J. Nieminen, Screens and rounding mappings, Zeitschrift f¨ur Angewandte Mathematik undMechanik 58 (1978), 519–520.
- [13] J. Nieminen, Blocks, error algebras and flou sets, Glasnik Matematički 14 (1979), 381–385.
- [14] J. Pogonowski, Tolerance spaces with applications to linguistics, University Press, Institute of Linguistics, Adam Mickiewicz University, Pozna´n, 1981.
- [15] Ju. A. Schreider, Equality, Resemblance, and Order, Mir Publishers, Moskow, 1975.
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- [17] E. C. Zeeman, The topology of the brain and visual perception, Topology of 3-Manifolds (M. K. Fort, ed.), Prentice-Hall, Englewood Cliffs, NJ, 1962.
- [18] B. Zelinka, Tolerance graphs, Commentationes Mathematicae Universitatis Carolinae 9 (1968), 121–131.
- [19] B. Zelinka, A remark on systems of maximal cliques of a graph, Czechoslovak Mathematical Journal 27 (1977), 617–618
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-8203dcf7-fa02-4678-b27b-a12aa408236b