Tolerances Induced by Irredundant Coverings

• Autorzy: Järvinen J. , Radeleczki S.

Identyfikatory

DOI

10.3233/FI-2015-1183

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

tolerance relation; irredundant covering; normal covering; clique of a graph; Helly number; rough set; formal concept analysis

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2015
• Tom: Vol. 137, nr 3
• Strony: 341--353
• Opis fizyczny: Bibliogr. 19 poz., rys.

Twórcy

autor

Järvinen J.

• Sirkankuja 1, 20810 Turku, Finland

autor

Radeleczki S.

• Institute of Mathematics, University of Miskolc 3515 Miskolc-Egyetemv´aros, Hungary

Bibliografia

• [1] J. Alt and N. Schofield, Clique analysis of a tolerance relation, The Journal ofMathematical Sociology 6 (1978), 155–162.
• [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.
• [4] I. Chajda, J. Niederle, and B. Zelinka, On existence conditions for compatible tolerances, Czechoslovak Mathematical Journal 26 (1976), 304–311.
• [5] G. Czédli, Factor lattices by tolerances, Acta Scientiarum Mathematicarum (Szeged) 44 (1982), 35–42.
• [6] G. Czédli and L. Klukovits, A note on tolerances of idempotent algebras, Glasnik Matemati˘cki 18 (1983), 35–38.
• [7] B. Ganter and R. Wille, Formal concept analysis: Mathematical foundations, Springer, Berlin/Heidelberg, 1999.
• [8] G. Grätzer and G. H.Wenzel, Tolerances, covering systems, and the axiom of choice, Archivum Mathematicum 25 (1989), 27–34.
• [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.
• [11] J. Järvinen and S. Radeleczki, Rough sets determined by tolerances, International Journal of Approximate Reasoning 55 (2014), 1419–1438.
• [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.
• [16] Y. Yao and B. Yao, Covering based rough set approximations, Information Sciences 200 (2012), 91–107.
• [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