Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T13:31:31.944Z Has data issue: false hasContentIssue false
  • English
  • Français

VARIETIES WHOSE TOLERANCES ARE HOMOMORPHIC IMAGES OF THEIR CONGRUENCES

Part of: Lattices Semigroups Algebraic structures

Published online by Cambridge University Press:  08 August 2012

GÁBOR CZÉDLI  and
EMIL W. KISS
GÁBOR CZÉDLI*
Affiliation:
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary (email: czedli@math.u-szeged.hu)
EMIL W. KISS
Affiliation:
Department of Algebra and Number Theory, Eötvös University, Pázmány Péter sétány 1/c, 1117 Budapest, Hungary (email: ewkiss@cs.elte.hu)
*
✉For correspondence; e-mail: czedli@math.u-szeged.hu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsev-like condition, we characterise varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras have TImC. We show that a congruence n-permutable variety has TImC if and only if it is congruence permutable, and construct an idempotent variety with a majority term that fails TImC.

Keywords

tolerance relationcongruence imagevariety of algebraslatticesunary algebrassemigroupsMalcev-like condition

MSC classification

Secondary: 08A30: Subalgebras, congruence relations 08A60: Unary algebras 06B10: Ideals, congruence relations 20M99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 2 , April 2013 , pp. 326 - 338
Copyright
©2012 Australian Mathematical Publishing Association Inc.

Footnotes

Dedicated to Béla Csákány on his eightieth birthday

References

[1]Chajda, I., ‘Partitions, coverings and blocks of compatible relations’, Glas. Mat. 14(34) (1979), 21–26.Google Scholar
[2]Chajda, I., Algebraic Theory of Tolerance Relations (Palacký University Olomouc, Olomouc, 1991).Google Scholar
[3]Chajda, I. & Czédli, G., ‘How to generate the involution lattice of quasiorders?’, Stud. Sci. Math. Hungar. 32 (1996), 415–427.Google Scholar
[4]Chajda, I., Czédli, G. & Halaš, R., ‘Independent joins of tolerance factorable varieties’. Algebra Universalis, to appear.Google Scholar
[5]Chajda, I., Czédli, G., Halaš, R. & Lipparini, P., ‘Tolerances as images of congruences in varieties defined by linear identities’, Algebra Universalis, to appear.Google Scholar
[6]Chajda, I., Niederle, J. & Zelinka, B., ‘On existence conditions for compatible tolerances’, Czech. Math. J. 26 (1976), 304–311.CrossRefGoogle Scholar
[7]Czédli, G., ‘Factor lattices by tolerances’, Acta Sci. Math. (Szeged) 44 (1982), 35–42.Google Scholar
[8]Czédli, G & Grätzer, G., ‘Lattice tolerances and congruences’, Algebra Universalis 66 (2011), 5–6.CrossRefGoogle Scholar
[9]Czédli, G., Horváth, E. K. & Lipparini, P., ‘Optimal Mal’tsev conditions for congruence modular varieties’, Algebra Universalis 53 (2005), 267–279.CrossRefGoogle Scholar
[10]Czédli, G. & Klukovits, L., ‘A note on tolerances of idempotent algebras’, Glas. Mat. 18(38) (1983), 35–38.Google Scholar
[11]Day, A. & Herrmann, Ch., ‘Gluings of modular lattices’, Order 5 (1988), 85–101.CrossRefGoogle Scholar
[12]Fried, E. & Grätzer, G., ‘Notes on tolerance relations of lattices: a conjecture of R. N. McKenzie’, J. Pure Appl. Algebra 68 (1990), 127–134.CrossRefGoogle Scholar
[13]Hagemann, J. & Mitschke, A., ‘On n-permutable congruences’, Algebra Universalis 3 (1973), 8–12.CrossRefGoogle Scholar
[14]Herrmann, Ch., ‘S-verklebte Summen von Verbänden’, Math. Z. 130 (1973), 255–274.CrossRefGoogle Scholar
[15]Jónsson, B., ‘Algebras whose congruence lattices are distributive’, Math. Scand. 21 (1967), 110–121.CrossRefGoogle Scholar
[16]Kindermann, M., ‘Über die Äquivalenz von Ordnungspolynomvollständigkeit und Toleranzeinfachheit endlicher Verbände’, Proc. Klagenfurt Conf. 1978 (1979), 145–149.Google Scholar
[17]Mal’cev, A. I., ‘On the general theory of algebraic systems’, Mat. Sb. N. Ser. 35(77) (1954), 3–20 (in Russian).Google Scholar
[18]Mitschke, A., ‘Implication algebras are 3-permutable and 3-distributive’, Algebra Universalis 1 (1971), 182–186.CrossRefGoogle Scholar
[19]Schmidt, E. T., ‘Kongruenzrelationen algebraischer Strukturen’, Math. Forsch. 25 (1969), Berlin.Google Scholar
[20]Smith, J. D. H., Mal’cev Varieties (Springer, Berlin, 1976).CrossRefGoogle Scholar
[21]Taylor, W., ‘Characterizing Mal’cev conditions’, Algebra Universalis 3 (1973), 351–397.CrossRefGoogle Scholar
[22]Werner, H., ‘A Mal’cev condition for admissible relations’, Algebra Universalis 3 (1973), 263.CrossRefGoogle Scholar